An explicit classification of dual pairs in exceptional Lie algebras

Abstract.

The primary goal of this paper is to explicitly write down all semisimple dual pairs in the exceptional Lie algebras. (A dual pair in a reductive Lie algebra $\mathfrak{g}$ is a pair of subalgebras such that each member equals the other’s centralizer in $\mathfrak{g}$ .) In a 1994 paper, H. Rubenthaler outlined a process for generating a complete list of candidate dual pairs in each of the exceptional Lie algebras. However, the process of checking whether each of these candidate dual pairs is in fact a dual pair is not easy, and requires several distinct insights and methods. In this paper, we carry out this process and explain the relevant concepts as we go. We also give plenty of examples with the hopes of making Rubenthaler’s 1994 result not only more complete but more usable and understandable.

The author was supported by the NSF Graduate Research Fellowship Program under Grant No. 2141064, and by the Fannie & John Hertz Foundation.

1. Introduction

The notion of a reductive dual pair¹¹1Italicized terms will be defined later in the paper. in a complex reductive Lie algebra was first introduced by Howe in his seminal 1989 paper [9], in which he also classified the reductive dual pairs in the symplectic Lie algebra $\mathfrak{sp}_{2m}$ . Since then, reductive dual pairs have been widely used and studied. For example, certain reductive dual pairs are implicitly used in some of the well-known constructions of the exceptional Lie algebras: $(G_{2}^{1},A_{2}^{2^{\prime\prime}})$ and $(G_{2}^{1},F_{4}^{1})$ correspond to the Freudenthal-Tits constructions of $E_{6}$ and $E_{8}$ , respectively [17], and $(D_{4},D_{4})$ corresponds to the triality construction of $E_{8}$ [1, 2]. Additionally, reductive dual pairs have some beautiful connections to the study of commuting nilpotent elements in complex semisimple Lie algebras as introduced by Ginzburg in [7]; see [14] for an explanation of these connections. More recently, the real forms of certain reductive dual pairs in exceptional Lie algebras have also drawn some attention [11, 12].

Some of the earliest work on reductive dual pairs appears in Rubenthaler’s 1994 paper [15]. In this paper, Rubenthaler outlines a classification of reductive dual pairs in complex reductive Lie algebras. While Rubenthaler’s paper provides a helpful framework, writing down all of these pairs explicitly requires a lot of additional nontrivial work. The primary goal of this paper is to explicitly write down all of the semisimple dual pairs in the exceptional Lie algebras. Here is how we will proceed:

•

Section 2: We reduce the problem of classifying reductive dual pairs in reductive Lie algebras to the problem of classifying semisimple dual pairs in simple Lie algebras.
•

Section 3: We classify dual pairs coming from maximal regular subalgebras of maximal rank.
•

Section 4: We define and classify admissible subalgebras.
•

Section 5: We introduce $S$ -irreducible dual pairs and admissible dual pairs, and describe how they nearly all arise from admissible subalgebras.
•

Section 6: We classify the semisimple dual pairs in the complex classical simple Lie algebras that are $S$ -irreducible, admissible, or that arise from maximal regular subalgebras of maximal rank. We write out all such dual pairs for rank up to 6.
•

Section 7: We explain how to obtain lists of candidate non- $S$ -irreducible dual pairs (including all actual non- $S$ -irreducible dual pairs).
•

Section 8: We introduce methods for determining whether or not a candidate dual pair is actually a dual pair.
•

Section 9: We classify all ( $S$ -irreducible and non- $S$ -irreducible) semisimple dual pairs in the exceptional Lie algebras.

2. Preliminaries

In this section, we introduce several important notions and conventions, and we explain why we are focusing on semisimple dual pairs in simple Lie algebras.

Definition 1 ([9, p. 550]).

Let $\mathfrak{g}$ be a reductive Lie algebra. Then a pair $(\mathfrak{a},\mathfrak{b})$ of subalgebras of $\mathfrak{g}$ is a reductive dual pair if

(1)

$\mathfrak{a}$ and $\mathfrak{b}$ are reductive in $\mathfrak{g}$ , and
(2)

$\mathfrak{a}=\mathfrak{z}_{\mathfrak{g}}(\mathfrak{b})$ and $\mathfrak{b}=\mathfrak{z}_{\mathfrak{g}}(\mathfrak{a})$ ,

where $\mathfrak{z}_{\mathfrak{g}}(\cdot)$ denotes taking the centralizer in $\mathfrak{g}$ .

Since we will only be considering reductive dual pairs in this paper, we will just use “dual pair” to refer to a reductive dual pair.

2.1. Why we are focusing on semisimple dual pairs in simple Lie algebras

As discussed in [15, Section 5], we can reduce the study of reductive dual pairs in reductive Lie algebras to the study of semisimple dual pairs in simple Lie algebras. In particular, the following result reduces the problem to finding reductive dual pairs in semisimple Lie algebras:

Lemma 2 ([15, Lemma 5.1]).

Let $\mathfrak{g}$ be a reductive Lie algebra. Write $\mathfrak{g}=Z_{\mathfrak{g}}\oplus[\mathfrak{g},\mathfrak{g}]$ (where $Z_{\mathfrak{g}}$ denotes the center of $\mathfrak{g}$ ). If $(\mathfrak{a},\mathfrak{b})$ is a dual pair in $\mathfrak{g}$ , then $\mathfrak{a}=Z_{\mathfrak{g}}\oplus\mathfrak{a}_{1}$ and $\mathfrak{b}=Z_{\mathfrak{g}}\oplus\mathfrak{b}_{1}$ , where $(\mathfrak{a}_{1},\mathfrak{b}_{1})$ is a dual pair in $[\mathfrak{g},\mathfrak{g}]$ . Conversely, any dual pair in $[\mathfrak{g},\mathfrak{g}]$ gives a dual pair of $\mathfrak{g}$ in this way.

Lemma 4 further shows that any reductive dual pair $(\mathfrak{a},\mathfrak{b})$ in a semisimple Lie algebra $\mathfrak{g}$ can be written in terms of $Z:=\mathfrak{a}\cap\mathfrak{b}$ and a semisimple dual pair in $[\ell_{Z},\ell_{Z}]$ , where $\ell_{Z}=\mathfrak{z}_{\mathfrak{g}}(Z)$ is a Levi subalgebra of $\mathfrak{g}$ with center $Z$ . Before formally stating this result, let us introduce Levi subalgebras (following [15, Section 1.2]):

Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$ . Let $\Psi=(\alpha_{1},\ldots,\alpha_{n})$ be a choice of simple roots associated to $(\mathfrak{g},\mathfrak{h})$ . For a subset $\theta$ of $\Psi$ , let $H_{\theta}$ be the unique element of $\mathfrak{h}$ defined by

\left\{\begin{array}[]{ll}\alpha(H_{\theta})=0&\text{if }\alpha\in\theta\\ \alpha(H_{\theta})=2&\text{if }\alpha\in\Psi\setminus\theta\end{array}\right.

For $p\in\mathbb{Z}$ , define

d_{p}(\theta)=\{X\in\mathfrak{g}\;|\;[H_{\theta},X]=2pX\}\hskip 14.22636pt\text{ so that }\hskip 14.22636pt\mathfrak{g}=\bigoplus_{p\in\mathbb{Z}}d_{p}(\theta).

Definition 3 ([15, Section 1.2]).

The reductive algebra $\ell_{\theta}:=d_{0}(\theta)$ is called the standard Levi subalgebra associated to $\theta$ .

Lemma 4 ([15, Lemma 5.2]).

Let $\mathfrak{g}$ be a semisimple Lie algebra and let $(\mathfrak{a},\mathfrak{b})$ be a dual pair in $\mathfrak{g}$ . If $\mathfrak{a}$ has nontrivial center $Z_{\mathfrak{a}}$ , then $\mathfrak{a}\cap\mathfrak{b}=Z_{\mathfrak{a}}=Z_{\mathfrak{b}}=:Z$ . Let $\ell_{Z}=\mathfrak{z}_{\mathfrak{g}}(Z)$ . The algebra $\ell_{Z}$ is a Levi subalgebra of $\mathfrak{g}$ whose center is $Z$ . If we write $\ell_{Z}=Z\oplus[\ell_{Z},\ell_{Z}]$ , $\mathfrak{a}=Z\oplus[\mathfrak{a},\mathfrak{a}]$ , and $\mathfrak{b}=Z\oplus[\mathfrak{b},\mathfrak{b}]$ , then $([\mathfrak{a},\mathfrak{a}],[\mathfrak{b},\mathfrak{b}])$ is a dual pair in $[\ell_{Z},\ell_{Z}]$ . Conversely, any dual pair in $[\ell_{Z},\ell_{Z}]$ gives a dual pair of $\mathfrak{g}$ in this way.

Finally, the following result further reduces the problem to finding semisimple dual pairs in simple Lie algebras:

Lemma 5 ([15, Lemma 5.3]).

Let $\mathfrak{g}$ be a semisimple Lie algebra and let $\mathfrak{g}=\mathfrak{g}_{1}\oplus\cdots\oplus\mathfrak{g}_{n}$ be its decomposition into simple ideals. If for each $i\in\{1,\ldots,n\}$ we have a dual pair $(\mathfrak{a}_{i},\mathfrak{b}_{i})$ in $\mathfrak{g}_{i}$ , then $\left(\prod_{i=1}^{n}\mathfrak{a}_{i},\prod_{i=1}^{n}\mathfrak{b}_{i}\right)$ is a dual pair in $\mathfrak{g}$ . Conversely, any dual pair in $\mathfrak{g}$ is obtained in this way.

Note that we will often omit the adjective “semisimple” in the following discussion of the semisimple dual pairs in the complex simple Lie algebras. (All subalgebras that follow can be assumed to be semisimple.)

2.2. Defining a normalized symmetric form on $\mathfrak{g}$

Let $\mathfrak{g}$ be a complex simple Lie algebra, and fix a Cartan subalgebra $\mathfrak{h}$ . Let $\mathcal{R}$ denote the root system corresponding to $(\mathfrak{g},\mathfrak{h})$ . Let $\mathfrak{g}_{0}$ be a compact real form of $\mathfrak{g}$ with Cartan $\mathfrak{h}_{0}$ (defined over $\mathbb{R}$ ) satisfying $(\mathfrak{h}_{0})_{\mathbb{C}}=\mathfrak{h}$ . Recall that any $\alpha\in\mathcal{R}$ is determined (by $\mathbb{C}$ -linearity) by its values on $i\mathfrak{h}_{0}$ , on which it is real-valued. Therefore, we can view $\mathcal{R}$ as sitting inside $(i\mathfrak{h}_{0})^{*}$ . We identify $(i\mathfrak{h}_{0})^{*}$ with a subset of $\mathbb{R}^{n}$ such that the standard Euclidean inner product $\langle\cdot,\cdot\rangle$ on $\mathbb{R}^{n}$ restricts to a Weyl group-invariant form on $(i\mathfrak{h}_{0})^{*}$ . In particular, we use the root system realizations described in [3, Plates I–IX], writing elements of $i\mathfrak{h}_{0}$ in terms of the standard Euclidean basis $\{e_{1},\ldots,e_{n}\}$ with $e_{1}=(1,0,\ldots,0)$ , etc., and writing elements of $(i\mathfrak{h}_{0})^{*}$ in terms of the corresponding standard basis of linear functionals $\{\varepsilon_{1},\ldots,\varepsilon_{n}\}$ .

Following [5, Section 2], we normalize a non-degenerate invariant symmetric bilinear form $(,)_{\mathfrak{g}}$ on $\mathfrak{g}$ as follows: for $\beta^{\vee}$ a short coroot in $\mathcal{R}^{\vee}$ , we require that $(\beta^{\vee},\beta^{\vee})_{\mathfrak{g}}=2$ . We also define a non-degenerate invariant symmetric bilinear form $(,)_{\mathfrak{g}^{*}}$ on $\mathfrak{g}^{*}$ by the condition $(\beta,\beta)_{\mathfrak{g}^{*}}=2$ for $\beta$ a long root in $\mathcal{R}$ . (Note that $(,)_{\mathfrak{g}^{*}}$ can also be defined as the dual form to the form on $\mathfrak{g}$ ; this dual form automatically satisfies the aforementioned normalizing condition.)

2.3. Dynkin’s index notation

We now introduce Dynkin’s index notation (cf. [5, Section 2]). Let $\varphi:\mathfrak{a}\rightarrow\mathfrak{g}$ be a homomorphism of complex simple Lie algebras. For $x,y\in\mathfrak{a}$ , the bilinear form $(x,y)\mapsto(\varphi(x),\varphi(y))_{\mathfrak{g}}$ is proportional to $(x,y)\mapsto(x,y)_{\mathfrak{a}}$ . The index $i_{\varphi}$ of $\varphi$ is defined by

(\varphi(x),\varphi(y))_{\mathfrak{g}}=i_{\varphi}\cdot(x,y)_{\mathfrak{a}},\hskip 7.11317ptx,y\in\mathfrak{a};

the index is always an integer [5, Theorem 2.2]. In particular, if $\mathfrak{a}$ is a simple subalgebra of $\mathfrak{g}$ , then the index of $\mathfrak{a}$ in $\mathfrak{g}$ is given by

i_{\mathfrak{a}\hookrightarrow\mathfrak{g}}=\frac{(x,x)_{\mathfrak{g}}}{(x,x)_{\mathfrak{a}}},\hskip 7.11317ptx\in\mathfrak{a}.

When we have a sequence of inclusions $\mathfrak{a}\hookrightarrow\mathfrak{a}^{\prime}\hookrightarrow\mathfrak{g}$ , the index satisfies the following multiplicative property:

i_{\mathfrak{a}\hookrightarrow\mathfrak{g}}=i_{\mathfrak{a}\hookrightarrow\mathfrak{a}^{\prime}}\cdot i_{\mathfrak{a}^{\prime}\hookrightarrow\mathfrak{g}}.

The index of $\mathfrak{a}$ in $\mathfrak{g}$ often determines the conjugacy class of $\mathfrak{a}$ (where the conjugation is by the adjoint group of $\mathfrak{g}$ ). When the conjugacy class of $\mathfrak{a}$ is fully specified by its type and index, we will follow Dynkin in writing $\text{Type}(\mathfrak{a})^{i_{\mathfrak{a}\hookrightarrow\mathfrak{g}}}$ to refer to the conjugacy class of $\mathfrak{a}$ . If, on the other hand, the conjugacy class of $\mathfrak{a}$ is not specified by its type and index, we will add “primes” to distinguish the possible conjugacy classes. For example, by [5, Table 25], there are two conjugacy classes of subalgebras of the complex Lie algebra $E_{7}$ that have type $A_{2}$ and index 2; we will refer to them as $A_{2}^{2^{\prime}}$ and $A_{2}^{2^{\prime\prime}}$ (by order of appearance in [5, Table 25]). Similarly, by Appendix B, there are two conjugacy classes of subalgebras of the complex Lie algebra $B_{4}$ that have type $B_{2}$ and index 1; we will refer to them as $B_{2}^{1^{\prime}}$ and $B_{2}^{1^{\prime\prime}}$ (by order of appearance in Table 19). To know whether we need to add these “primes,” we will continue to refer to [5] for subalgebras of the exceptional Lie algebras and to Appendices A and B for subalgebras of the classical simple Lie algebras.

Example 6.

Consider $\mathfrak{sp}_{2}\oplus\mathfrak{sp}_{2m}\subset\mathfrak{so}_{4m}$ . (Here, $\mathfrak{sp}_{2}\hookrightarrow\mathfrak{so}_{4m}$ can be thought of as a diagonal embedding with $2m$ copies, and $\mathfrak{sp}_{2m}\hookrightarrow\mathfrak{so}_{4m}$ can be thought of as “multiplying” each entry by the $2\times 2$ identity matrix.) Let’s compute the index of $\mathfrak{sp}_{2}$ and $\mathfrak{sp}_{2m}$ in $\mathfrak{so}_{4m}$ . Following [3, Plate IV], $\mathfrak{so}_{4m}$ has

\{\varepsilon_{1}-\varepsilon_{2},\,\varepsilon_{2}-\varepsilon_{3},\ldots,\,\varepsilon_{2m-1}-\varepsilon_{2m},\,\varepsilon_{2m-1}+\varepsilon_{2m}\}

as its simple roots, with corresponding simple coroots

\{e_{1}-e_{2},\,e_{2}-e_{3},\ldots,\,e_{2m-1}-e_{2m},\,e_{2m-1}+e_{2m}\}.

Set $z:=e_{1}-e_{2}$ . This coroot corresponds to the diagonal matrix

\text{diag}(1,-1,0,\ldots,0;-1,1,0,\ldots,0)

in $\mathfrak{so}_{4m}$ , so we see that the normalizing condition on $(\cdot,\cdot)_{\mathfrak{so}_{4m}}$ is

(z,z)_{\mathfrak{so}_{4m}}=2=\frac{1}{2}\langle z,z\rangle,

where $\langle\cdot,\cdot\rangle$ denotes the standard Euclidean inner product.

Now, following [3, Plate III], $\mathfrak{sp}_{2}$ has $2\varepsilon_{1}$ as its simple root, and has $x:=e_{1}$ as the corresponding simple coroot. Similarly, $\mathfrak{sp}_{2m}$ has simple roots

\{\varepsilon_{1}-\varepsilon_{2},\,\varepsilon_{2}-\varepsilon_{3},\ldots,\,\varepsilon_{m-1}-\varepsilon_{m},2\varepsilon_{m}\},

with corresponding simple coroots

\{e_{1}-e_{2},\,e_{2}-e_{3},\ldots,\,e_{m-1}-e_{m},\,y:=e_{m}\}.

Thus, the normalizing conditions on $(\cdot,\cdot)_{\mathfrak{sp}_{2}}$ and $(\cdot,\cdot)_{\mathfrak{sp}_{2m}}$ are $(x,x)_{\mathfrak{sp}_{2}}=2$ and $(y,y)_{\mathfrak{sp}_{2m}}=2$ .

Here, $x$ corresponds to the matrix $\text{diag}(1;-1)$ in $\mathfrak{sp}_{2}$ and $y$ corresponds to the matrix $\text{diag}(0,\ldots,0,1;0,\ldots,0,-1)$ in $\mathfrak{sp}_{2m}$ . Since $\mathfrak{sp}_{2}$ is embedded diagonally (with $2m$ copies) in $\mathfrak{so}_{4m}$ , we see that

	$\displaystyle(x,x)_{\mathfrak{so}_{4m}}=\frac{1}{2}\Big{\langle}$	$\displaystyle\text{diag}(1,-1,\ldots,1,-1;-1,1,\ldots,-1,1),$
		$\displaystyle\text{diag}(1,-1,\ldots,1,-1;-1,1,\ldots,-1,1)\Big{\rangle}=2m.$

Similarly, we see that

	$\displaystyle\displaystyle(y,y)_{\mathfrak{so}_{4m}}=\frac{1}{2}\Big{\langle}$	$\displaystyle\text{diag}(0,\ldots,0,1,1;0,\ldots,0,-1,-1),$
		$\displaystyle\text{diag}(0,\ldots,0,1,1;0,\ldots,0,-1,-1)\Big{\rangle}=2.$

It follows that

i_{\mathfrak{sp}_{2}\hookrightarrow\mathfrak{so}_{4m}}=\frac{(x,x)_{\mathfrak{so}_{4m}}}{(x,x)_{\mathfrak{sp}_{2}}}=\frac{2m}{2}=m\hskip 14.22636pt\text{ and }\hskip 14.22636pti_{\mathfrak{sp}_{2m}\hookrightarrow\mathfrak{so}_{4m}}=\frac{(y,y)_{\mathfrak{so}_{4m}}}{(y,y)_{\mathfrak{sp}_{2m}}}=\frac{2}{2}=1.

In this way, we see that $\mathfrak{sp}_{2}=C_{1}^{m}\simeq A_{1}^{m}$ (possibly with “primes”) and that $\mathfrak{sp}_{2m}=C_{m}^{1}$ (possibly with “primes”). This will appear in later examples of dual pairs; specifically, $(A_{1}^{2^{\prime}},B_{2}^{1^{\prime}})$ , $(A_{1}^{2^{\prime\prime}},B_{2}^{1^{\prime\prime}})$ , and $(A_{1}^{2^{\prime\prime\prime}},B_{2}^{1^{\prime\prime\prime}})$ are dual pairs in $D_{4}$ , and $(A_{1}^{3^{\prime}},C_{3}^{1^{\prime\prime}})$ and $(A_{1}^{3^{\prime\prime}},C_{3}^{1^{\prime}})$ are dual pairs in $D_{6}$ (see Table 7).

2.4. Dynkin’s defining vector conventions and weighted diagrams

Let $\mathfrak{a}$ be a three-dimensional semisimple subalgebra of a complex simple Lie algebra $\mathfrak{g}$ . (Any such subalgebra is isomorphic to $\mathfrak{sl}_{2}$ .) A vector $H\in\mathfrak{a}$ which can be supplemented with two vectors $X,Y\in\mathfrak{a}$ to form an $\mathfrak{sl}_{2}$ -triplet, i.e. a basis of $\mathfrak{a}$ satisfying the commutator relations

	$\displaystyle[H,X]$	$\displaystyle=2X$
	$\displaystyle[H,Y]$	$\displaystyle=-2Y$
	$\displaystyle[X,Y]$	$\displaystyle=H,$

is called a defining vector of $\mathfrak{a}$ .

Remark 7.

For a three-dimensional subalgebra $\mathfrak{a}$ of $\mathfrak{g}$ with defining vector $H$ , the index can be computed as

i_{\mathfrak{a}\hookrightarrow\mathfrak{g}}=\frac{(H,H)_{\mathfrak{g}}}{(H,H)_{\mathfrak{a}}}=\frac{(H,H)_{\mathfrak{g}}}{2}.

Fix a defining vector $H$ for $\mathfrak{a}$ . Choose a Cartan subalgebra $\mathfrak{h}$ containing $H$ , as well as a positive root system making $H$ weakly dominant (i.e. $\alpha(H)\geq 0$ whenever $\alpha>0$ ). Let $\Pi$ be the corresponding system of simple roots. In [5], Dynkin often writes the defining vector of $\mathfrak{a}$ in terms of the simple coroots of $\mathfrak{g}$ (appropriately normalized). In particular, let $\Pi_{\ell}$ denote the set of long simple roots in $\Pi$ and let $\Pi_{s}$ denote the set of short simple roots in $\Pi$ . Set $r:=(\beta,\beta)_{\mathfrak{g}}/(\gamma,\gamma)_{\mathfrak{g}}$ for $\beta\in\Pi_{\ell}$ and $\gamma\in\Pi_{s}$ (so that $r=1$ if $\mathfrak{g}$ has only one root length, $r=3$ if $\mathfrak{g}$ is of type $G_{2}$ , and $r=2$ in all other cases of two root lengths). Then Dynkin writes the defining vector $H$ of $\mathfrak{a}$ as

H=\sum_{\beta\in\Pi_{\ell}}a_{\beta}\beta^{\vee}+\sum_{\gamma\in\Pi_{s}}a_{\gamma}\frac{\gamma^{\vee}}{r},

and he calls the coefficients $\{a_{\beta},a_{\gamma}\}_{\beta\in\Pi_{\ell},\,\gamma\in\Pi_{s}}$ the coordinates of the defining vector $H$ .

Now, consider the Dynkin diagram for $\mathfrak{g}$ . There is a unique correspondence between the nodes of this diagram and the elements of $\Pi$ . Let us associate the number $\alpha(H)$ with the node corresponding to $\alpha$ . The resulting diagram (with numbers written in) is called the weighted diagram of the subalgebra $\mathfrak{a}$ .

Figure 1. Root labeling conventions for types

G_{2}

F_{4}

E_{6}

E_{7}

, and

E_{8}

Example 8.

To illustrate the above definitions, let us verify some of the entries in [5, Table 16], which states the index, weighted diagram, and defining vector for the three-dimensional subalgebras of the exceptional Lie algebra $\mathfrak{g}$ of type $G_{2}$ .

By [3, Plate IX], we can realize this root system as the hyperplane in $\mathbb{R}^{3}$ defined by the equation $\xi_{1}+\xi_{2}+\xi_{3}=0$ , with the following roots:

		$\displaystyle\pm(\varepsilon_{1}-\varepsilon_{2}),\hskip 7.11317pt\pm(\varepsilon_{1}-\varepsilon_{3}),\hskip 7.11317pt\pm(\varepsilon_{2}-\varepsilon_{3}),$
	$\displaystyle\pm(2\varepsilon_{1}$	$\displaystyle-\varepsilon_{2}-\varepsilon_{3}),\hskip 7.11317pt\pm(2\varepsilon_{2}-\varepsilon_{1}-\varepsilon_{3}),\hskip 7.11317pt\pm(2\varepsilon_{3}-\varepsilon_{1}-\varepsilon_{2}).$

This system has basis $\alpha_{1}=-2\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}$ and $\alpha_{2}=\varepsilon_{1}-\varepsilon_{2}$ (cf. Figure 1), with corresponding coroots $\alpha_{1}^{\vee}=\frac{1}{3}(-2,1,1)$ and $\alpha_{2}^{\vee}=(1,-1,0)$ .

Let us consider the first subalgebra in [5, Table 16], with defining vector coordinates $(2,3)$ . Then the defining vector is given by

H:=2\alpha_{1}^{\vee}+3\left(\frac{\alpha_{2}^{\vee}}{3}\right)=\left(-\frac{1}{3},-\frac{1}{3},\frac{2}{3}\right).

So we see that

	$\displaystyle\alpha_{1}(H)$	$\displaystyle=\frac{2}{3}-\frac{1}{3}+\frac{2}{3}=1,\text{ and }$
	$\displaystyle\alpha_{2}(H)$	$\displaystyle=-\frac{1}{3}+\frac{1}{3}+0=0.$

Therefore, the weighted diagram of this three-dimensional subalgebra is

as expected. Since $\alpha_{1}^{\vee}$ is our short coroot, the normalizing condition on $(\cdot,\cdot)_{\mathfrak{g}}$ is that $(\alpha_{1}^{\vee},\alpha_{1}^{\vee})_{\mathfrak{g}}=2=3\cdot\langle\alpha_{1}^{\vee},\alpha_{1}^{\vee}\rangle$ . Therefore, the index of this subalgebra in $G_{2}$ is given by

\frac{(H,H)_{\mathfrak{g}}}{2}=\frac{3\cdot(2/3)}{2}=1,

as expected.

3. Maximal and maximal-rank regular subalgebras

In this section, we introduce the notions of maximal regular subalgebras and maximal-rank regular subalgebras. We also classify the semisimple dual pairs coming from maximal regular subalgebras of maximal rank (see Proposition 15 and Table 3) and discuss briefly how maximal-rank regular subalgebras will play a crucial role later in the paper.

Definition 9 ([15, Definition 5.6]).

Let $\mathfrak{g}$ be a semisimple Lie algebra. A subalgebra $\mathfrak{u}$ of $\mathfrak{g}$ is called regular if there exists a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ such that $\mathfrak{u}$ is invariant under $\text{ad}_{\mathfrak{g}}\mathfrak{h}$ (i.e. $[\mathfrak{h},\mathfrak{u}]\subset\mathfrak{u}$ ).

Definition 10.

Let $\mathfrak{g}$ be a semisimple Lie algebra.

•

A maximal regular subalgebra of $\mathfrak{g}$ is a proper regular semisimple subalgebra $\widetilde{\mathfrak{g}}$ of $\mathfrak{g}$ for which no regular semisimple subalgebra $\mathfrak{g}^{*}$ exists such that $\widetilde{\mathfrak{g}}\subsetneq\mathfrak{g}^{*}\subsetneq\mathfrak{g}$ .
•

A maximal-rank regular subalgebra of $\mathfrak{g}$ is a proper regular semisimple subalgebra with rank equal to the rank of $\mathfrak{g}$ .²²2In [5] and [15], Dynkin and Rubenthalter do not require maximal regular subalgebras or maximal-rank regular subalgebras to be semisimple. Since the maximal/maximal-rank regular subalgebras we consider in this paper will all be semisimple, we include the “semisimple” qualifier in the definition for convenience.

As we will see below, a maximal-rank regular subalgebra need not be maximal, and a maximal regular subalgebra need not have maximal rank. It will be important to keep this in mind moving forward. In fact, mistakenly conflating these notions seems to have contributed to some errors in [5] and [15] that we will soon be discussing.

3.1. Maximal-rank regular subalgebras

Let $\mathfrak{g}$ be a simple Lie algebra. It was shown in [5, No. 17] that the maximal-rank regular subalgebras in $\mathfrak{g}$ are obtained (up to conjugation) by applying a finite number of elementary operations to the Dynkin diagram $D$ of $\mathfrak{g}$ .

In each elementary operation, we first pass from $D$ to the extended Dynkin diagram $\widetilde{D}$ of $\mathfrak{g}$ , which is the diagram obtained from $D$ by adding a node corresponding to $\widetilde{\alpha}$ (where $\widetilde{\alpha}$ is the lowest root of $\mathfrak{g}$ ). This new node gets connected to $D$ using the usual Dynkin diagram edge rules. The resulting diagram $\widetilde{D}$ is not the Dynkin diagram of a semisimple algebra, but if we remove a node $\alpha_{k}$ from $\widetilde{D}$ , we obtain the Dynkin diagram $D_{\alpha_{k}}$ of a semisimple algebra. When $\alpha_{k}=\widetilde{\alpha}$ , we simply recover the Dynkin diagram $D$ of $\mathfrak{g}$ . When $\alpha_{k}\neq\widetilde{\alpha}$ , it is not hard to check that the corresponding semisimple algebra, $\mathfrak{g}(\alpha_{k})$ (i.e. the subalgebra of $\mathfrak{g}$ generated by the root spaces $\mathfrak{g}^{\pm\alpha}$ for $\alpha\in D_{\alpha_{k}}$ ), is a maximal-rank regular subalgebra of $\mathfrak{g}$ . For $\alpha_{k}\neq\widetilde{\alpha}$ , the process of passing from $\mathfrak{g}$ to $\mathfrak{g}(\alpha_{k})$ that we have just described is called an elementary operation. For $\mathfrak{g}$ a semisimple Lie algebra, an elementary operation is the process of applying the above process to a single simple factor of $\mathfrak{g}$ .

Remark 11.

It is not hard to see from this analysis that the simple factors of maximal-rank regular subalgebras almost always have index 1. As a result, we will often omit the index of these subalgebras for convenience. The exceptions to this rule are the subalgebras of rank 1 coming from a short root in $B_{n}$ , $G_{2}$ , or $F_{4}$ (which have index 2, 3, and 2, respectively), and the subalgebra of type $A_{2}$ coming from the short roots in $F_{4}$ (which has index 2). We will follow the convention of denoting these exceptions with a tilde (e.g. $\widetilde{A_{2}}$ denotes the aforementioned index-2 subalgebra of $F_{4}$ ).

Example 12.

Consider the extended Dynkin diagram $\widetilde{D}$ for $\mathfrak{g}=E_{6}$ shown on the left (where the node corresponding to $\widetilde{\alpha}$ is indicated with a triangle). Crossing out the node corresponding to $\alpha_{2}$ , we obtain the diagram $D_{\alpha_{2}}$ shown on the right, which has type $A_{5}\oplus A_{1}$ . This corresponds to the semisimple algebra $\mathfrak{g}(\alpha_{2})=A_{5}\oplus A_{1}$ of $E_{6}$ (where both $A_{5}$ and $A_{1}$ have index 1 in $E_{6}$ ). This is a maximal-rank regular subalgebra of $E_{6}$ , which in fact turns out to be maximal regular as well.

Carrying out finitely many elementary operations like this, one can verify Table 1, which contains complete lists of the maximal-rank regular subalgebras of the classical simple and exceptional Lie algebras.

$\mathfrak{g}$	Subalgebra
$B_{n}$	$B_{m_{0}}\oplus D_{m_{1}}\oplus\cdots\oplus D_{m_{r}}$	( $r\geq 1$ , $m_{1}\geq\cdots\geq m_{r}>1$ , $\sum_{i}m_{i}=n$ )
$C_{n}$	$C_{\ell_{1}}\oplus\cdots\oplus C_{\ell_{r}}$	( $r\geq 2$ , $\ell_{1}\geq\cdots\geq\ell_{r}>0$ , $\sum_{i}\ell_{i}=n$ )
$D_{n}$	$D_{m_{1}}\oplus\cdots\oplus D_{m_{r}}$	( $r\geq 2$ , $m_{1}\geq\cdots\geq m_{r}>1$ , $\sum_{i}m_{i}=n$ )

$G_{2}$	$F_{4}$	$E_{6}$	$E_{7}$
$A_{2}$	$B_{4}$	$A_{5}\oplus A_{1}$	$D_{6}\oplus A_{1}$	$A_{8}$	$E_{7}\oplus A_{1}$
$A_{1}\oplus\widetilde{A_{1}}$	$A_{3}\oplus\widetilde{A_{1}}$	$3A_{2}$	$A_{5}^{{}^{\prime\prime}}\oplus A_{2}$	$D_{8}$	$D_{6}\oplus 2A_{1}$
	$A_{2}\oplus\widetilde{A_{2}}$		$2A_{3}\oplus A_{1}$	$A_{7}^{{}^{\prime}}\oplus A_{1}$	$D_{5}\oplus A_{3}$
	$C_{3}\oplus A_{1}$		$A_{7}$	$A_{5}\oplus A_{2}\oplus A_{1}$	$2D_{4}$
	$D_{4}$		$D_{4}\oplus 3A_{1}$	$2A_{4}$	$D_{4}\oplus 4A_{1}$
	$B_{2}\oplus 2A_{1}$		$7A_{1}$	$4A_{2}$	$2A_{3}\oplus 2A_{1}$
	$4A_{1}$			$E_{6}\oplus A_{2}$	$8A_{1}$

Table 1. Maximal-rank regular subalgebras of the simple Lie algebras (cf. [5, Tables 9, 11]).

Note that there are two conjugacy classes of subalgebras with index 1 and type $A_{5}$ in $E_{7}$ , as well as two conjugacy classes of subalgebras with index 1 and type $A_{7}$ in $E_{8}$ . However, there is a unique conjugacy class in $E_{7}$ with type $A_{5}\oplus A_{2}$ and with both factors having index 1. Similarly, there is a unique conjugacy class in $E_{8}$ with type $A_{7}\oplus A_{1}$ and with both factors having index 1. In these cases, we still include “primes” to clarify which conjugacy classes $A_{5}$ and $A_{7}$ belong to: $A_{5}^{{}^{\prime\prime}}\oplus A_{2}$ and $A_{7}^{{}^{\prime}}\oplus A_{1}$ [5, Table 25].

Note also that for certain values of $n$ , $m_{i}$ , and $\ell_{i}$ as above, there are multiple conjugacy classes of subalgebras in $B_{n}$ , $C_{n}$ , or $D_{n}$ with index 1 and type $B_{m_{0}}$ , $D_{m_{i}}$ , or $C_{\ell_{i}}$ . However, as in the exceptional case, the conjugacy class of each maximal-rank regular subalgebra is uniquely determined by type and index [5, No. 17]. While “primes” on simple factors are omitted from the classical portion of Table 1, we include them when describing the dual pairs in the classical simple Lie algebras of low rank in Section 6.

3.2. Maximal regular subalgebras of maximal rank

Some of the maximal-rank regular subalgebras shown in Table 1 are in fact maximal regular as well. By [5, No. 17], the regular subalgebras that are both maximal regular and maximal-rank are all obtained by means of only one elementary operation. Moreover, all of the subalgebras obtained by a single elementary operation are maximal-rank regular subalgebras, and most of them are maximal. (Note that Dynkin in [5, Theorem 5.5] mistakenly suggests that all maximal-rank regular subalgebras that are obtained by a single elementary operation are maximal. This seemingly led Dynkin to mistakenly include several non-maximal subalgebras in [5, Table 12]; in particular, he mistakenly includes $A_{3}\oplus\widetilde{A_{1}}$ as maximal in $F_{4}$ , includes $2A_{3}\oplus A_{1}$ as maximal in $E_{7}$ , and includes $A_{7}\oplus A_{1}$ , $A_{5}\oplus A_{2}\oplus A_{1}$ , and $A_{5}\oplus A_{3}$ as maximal in $E_{8}$ .)³³3These errors in [5, Table 12] were also noted in [16, Section 3]. Fortunately, there is a straightforward way to check whether a maximal-rank regular subalgebra that is obtained via a single elementary operation is maximal regular as well:

Lemma 13.

Let $\mathfrak{g}$ be a simple Lie algebra, $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$ , and $\{\alpha_{1},\ldots,\alpha_{n}\}$ the set of simple roots corresponding to $(\mathfrak{g},\mathfrak{h})$ . Consider the extended Dynkin diagram of $\mathfrak{g}$ with the node corresponding to $\alpha_{k}$ labeled with the coefficient $m_{k}$ of $\alpha_{k}$ in the highest root $\delta$ , and the node corresponding to the lowest root $\widetilde{\alpha}$ labeled 1. The regular subalgebras of $\mathfrak{g}$ that are both maximal regular and maximal-rank are exactly the subalgebras $\mathfrak{g}(\alpha_{k})$ with $m_{k}$ prime.

Proof.

By [5, No. 17], the maximal regular subalgebras of maximal rank are all obtained by means of a single elementary operation, and any subalgebra obtained by elementary operations is a maximal-rank regular subalgebra. Therefore, it remains to show that for a simple root $\alpha_{k}$ , $\mathfrak{g}(\alpha_{k})$ is maximal if and only if $m_{k}$ is prime.

To this end, fix $\alpha_{k}$ and let $\xi_{k}$ denote the corresponding fundamental coweight (so that $\xi_{k}(\alpha_{k})=1$ and $\xi_{k}(\alpha_{j})=0$ for $j\neq k$ ). Set

t_{k}:=\exp\left(\frac{2\pi i}{m_{k}}\xi_{k}\right).

Then $t_{k}$ acts on $\mathfrak{g}$ via the adjoint action, and hence defines an automorphism of $\mathfrak{g}$ satisfying $t_{k}^{m_{k}}=1$ . We claim that $\mathfrak{z}_{\mathfrak{g}}(t_{k})=\mathfrak{g}(\alpha_{k})$ . It suffices to show that $\{\widetilde{\alpha}\}\cup\{\alpha_{j}\}_{j\neq k}$ is the system of simple roots corresponding to $(\mathfrak{z}_{\mathfrak{g}}(t_{k}),\mathfrak{h})$ .

To start, note that $\alpha_{j}(t_{k})=1$ for any $j\neq k$ . Additionally, note that $\alpha_{k}(t_{k})=e^{\frac{2\pi i}{m_{k}}}$ , which gives that $\widetilde{\alpha}(t_{k})=\alpha_{k}(t_{k})^{m_{k}}=e^{2\pi i}=1$ . From this, it’s clear that $\mathfrak{g}^{\pm\widetilde{\alpha}}$ and $\mathfrak{g}^{\pm\alpha_{j}}$ ( $j\neq k$ ) are contained in $\mathfrak{z}_{\mathfrak{g}}(t_{k})$ . For the other direction, suppose we have some $\beta=\sum_{j=1}^{n}c_{j}\alpha_{j}$ in $\mathfrak{z}_{\mathfrak{g}}(t_{k})$ (where $-m_{j}\leq c_{j}\leq m_{j}$ and where either $c_{j}\leq 0$ for all $j$ or $c_{j}\geq 0$ for all $j$ ). Then

\beta(t_{k})=e^{\frac{2\pi ic_{k}}{m_{k}}}=1,

meaning $\frac{c_{k}}{m_{k}}\in\mathbb{Z}$ . Since $-m_{k}\leq c_{k}\leq m_{k}$ , we see that $c_{k}\in\{-m_{k},0,m_{k}\}$ . If $c_{k}=m_{k}$ , then $c_{j}\geq 0$ for all $j$ and

\beta=\sum_{j}c_{j}\alpha_{j}=-\widetilde{\alpha}-\sum_{j\neq k}(m_{j}-c_{j})\alpha_{j}

is a non-positive linear combination of our proposed set of simple roots. Similarly, if $c_{k}=-m_{k}$ , then $c_{j}\leq 0$ for all $j$ and

\beta=\sum_{j}c_{j}\alpha_{j}=\widetilde{\alpha}+\sum_{j\neq k}(m_{j}+c_{j})\alpha_{j}

is a non-negative linear combination of our proposed set of simple roots. Finally, if $c_{k}=0$ , there is nothing to show. It follows that $\mathfrak{z}_{\mathfrak{g}}(t_{k})=\mathfrak{g}(\alpha_{k})$ , as desired.

Now, suppose that $m_{k}$ is prime. We would like to show that $\mathfrak{z}_{\mathfrak{g}}(t_{k})=\mathfrak{g}(\alpha_{k})$ is a maximal regular subalgebra. To this end, note that $t_{k}$ induces a $(\mathbb{Z}/m_{k}\mathbb{Z})$ -grading on $\mathfrak{g}$ , where if $\beta(t_{k})=e^{\frac{2\pi ir}{m_{k}}}$ (with $0\leq r\leq m_{k}-1$ ), then $\mathfrak{g}^{\beta}$ is in the $r$ -th level of the grading. Let’s call these levels $\mathfrak{g}_{0},\ldots,\mathfrak{g}_{m_{k}-1}$ . It’s clear that $\mathfrak{g}_{0}=\mathfrak{z}_{\mathfrak{g}}(t_{k})=\mathfrak{g}(\alpha_{k})$ , and hence that each level is a module under the adjoint action of $\mathfrak{g}(\alpha_{k})$ . Suppose we can show that each level is irreducible under this action. Then any subalgebra $\mathfrak{r}$ of $\mathfrak{g}$ properly containing $\mathfrak{g}_{0}$ can be written as a sum of levels. Moreover, by the irreducibility of the levels, $[\mathfrak{g}_{i},\mathfrak{g}_{j}]=\mathfrak{g}_{i+j\pmod{m_{k}}}$ . Since $m_{k}$ is prime, closure under Lie bracket implies that $\mathfrak{r}=\mathfrak{g}$ .

Thus, in the case of $m_{k}$ prime, we have reduced to showing that the levels of the $(\mathbb{Z}/m_{k}\mathbb{Z})$ -grading on $\mathfrak{g}$ induced by $t_{k}$ are irreducible as $\mathfrak{g}(\alpha_{k})$ -modules. This is most straightforwardly proven by considering each case individually; we will not write out all of these cases, but instead will include an illustrative example. To this end, consider $\mathfrak{g}(\alpha_{5})$ in the case where $\mathfrak{g}$ is of type $E_{8}$ (where $\alpha_{5}$ is as in Figure 1). In this case, the extended Dynkin diagram with labels as described in the lemma statement is as follows:

From this, we see that $m_{5}=5$ and that the root vector $X_{\alpha_{5}}$ is the lowest weight for the irreducible representation $\wedge^{2}\mathbb{C}^{5}\otimes\mathbb{C}^{5}$ of $\mathfrak{g}(\alpha_{5})$ . It follows that $\mathfrak{g}_{1}$ has dimension at least $\binom{5}{2}\cdot 5=50$ . Now, since $t_{5}$ represents the unique conjugacy class of elements of order 5 with centralizer $\mathfrak{g}(\alpha_{5})$ – and since $t_{5}^{2}$ , $t_{5}^{3}$ , and $t_{5}^{4}$ have the same property – we see that $t_{5}$ , $t_{5}^{2}$ , $t_{5}^{3}$ , and $t_{5}^{4}$ are all conjugate. It follows that $\mathfrak{g}_{2}$ , $\mathfrak{g}_{3}$ , and $\mathfrak{g}_{4}$ also have irreducible components of dimension at least 50. Additionally, $\mathfrak{g}_{0}=\mathfrak{g}(\alpha_{5})=\mathfrak{sl}_{5}\oplus\mathfrak{sl}_{5}$ has dimension 48. Since $50\cdot 4+48=248=\dim\mathfrak{g}$ , this shows that these containments are in fact equalities, and hence that $\mathfrak{g}_{1}$ , $\mathfrak{g}_{2}$ , $\mathfrak{g}_{3}$ , and $\mathfrak{g}_{4}$ are irreducible, as desired.

Finally, suppose that $m_{k}$ is not prime, and that $p\mid m_{k}$ . It is clear that $\mathfrak{g}(\alpha_{k})=\mathfrak{z}_{\mathfrak{g}}(t_{k})\subseteq\mathfrak{z}_{\mathfrak{g}}(t_{k}^{p})$ . Moreover, taking $\frac{m_{k}}{p}$ as the coefficient of $\alpha_{k}$ , it is not difficult to construct roots of $\mathfrak{z}_{\mathfrak{g}}(t_{k}^{p})$ that cannot be written as non-positive or non-negative linear combinations of $\{\widetilde{\alpha}\}\cup\{\alpha_{j}\}_{j\neq k}$ . It follows that $\mathfrak{z}_{\mathfrak{g}}(t_{k})\subsetneq\mathfrak{z}_{\mathfrak{g}}(t_{k}^{p})$ . Additionally, $\mathfrak{z}_{\mathfrak{g}}(t_{k}^{p})$ is clearly regular. Therefore, $\mathfrak{g}(\alpha_{k})$ is not a maximal regular subalgebra in this case. ∎

Example 14.

For example, here is the extended diagram for $E_{7}$ , which has highest root $\delta=2\alpha_{1}+2\alpha_{2}+3\alpha_{3}+4\alpha_{4}+3\alpha_{5}+2\alpha_{6}+\alpha_{7}$ :

In this case, $\mathfrak{g}(\alpha_{1})=\mathfrak{g}(\alpha_{6})=A_{1}\oplus D_{6}$ , which is maximal by Lemma 13 (since the relevant elementary operation involves removing a node with prime label 2). Similarly, Lemma 13 gives that $\mathfrak{g}(\alpha_{3})=\mathfrak{g}(\alpha_{5})=A_{2}\oplus A_{5}^{{}^{\prime\prime}}$ is maximal. Removing either node with label 1, we get $\mathfrak{g}(\widetilde{\alpha})=\mathfrak{g}(\alpha_{7})=E_{7}$ , which is not a maximal regular subalgebra (since maximal regular subalgebras are defined to be proper). Finally, removing the node with label 4, we get $\mathfrak{g}(\alpha_{4})=A_{1}\oplus A_{3}\oplus A_{3}$ , which Lemma 13 says is not maximal regular. Indeed, since $A_{3}\simeq D_{3}$ , we see from Table 2 that $A_{1}\oplus A_{3}\oplus A_{3}\simeq A_{1}\oplus D_{3}\oplus D_{3}\subset A_{1}\oplus D_{6}$ .

Computing $\mathfrak{g}(\alpha_{k})$ for simple $\mathfrak{g}$ and simple roots $\alpha_{k}$ with prime label, we obtain the list of maximal regular subalgebras of maximal rank shown in Table 2.

$\mathfrak{g}$	Subalgebra	$\mathfrak{g}$	Subalgebra
$B_{n}$	$\mathfrak{g}(\alpha_{k})=D_{k}\oplus B_{n-k}$	$E_{6}$	$\mathfrak{g}(\alpha_{2})\simeq\mathfrak{g}(\alpha_{3})\simeq\mathfrak{g}(\alpha_{5})=A_{1}\oplus A_{5}$
	( $k=2,3,\ldots,n$ )		$\mathfrak{g}(\alpha_{4})=3A_{2}$
$C_{n}$	$\mathfrak{g}(\alpha_{k})=C_{k}\oplus C_{n-k}$	$E_{7}$	$\mathfrak{g}(\alpha_{1})\simeq\mathfrak{g}(\alpha_{6})=A_{1}\oplus D_{6}$
	( $k=1,2,\ldots,n$ )		$\mathfrak{g}(\alpha_{2})=A_{7}$
$D_{n}$	$\mathfrak{g}(\alpha_{k})=D_{k}\oplus D_{n-k}$		$\mathfrak{g}(\alpha_{3})\simeq\mathfrak{g}(\alpha_{5})=A_{2}\oplus A_{5}^{{}^{\prime\prime}}$
	( $k=2,3,\ldots,n-2$ )	$E_{8}$	$\mathfrak{g}(\alpha_{1})=D_{8}$
$F_{4}$	$\mathfrak{g}(\alpha_{1})=A_{1}\oplus C_{3}$		$\mathfrak{g}(\alpha_{2})=A_{8}$
	$\mathfrak{g}(\alpha_{2})=A_{2}\oplus\widetilde{A_{2}}$		$\mathfrak{g}(\alpha_{5})=2A_{4}$
	$\mathfrak{g}(\alpha_{4})=B_{4}$		$\mathfrak{g}(\alpha_{7})=A_{2}\oplus E_{6}$
$G_{2}$	$\mathfrak{g}(\alpha_{1})=A_{1}\oplus\widetilde{A_{1}}$		$\mathfrak{g}(\alpha_{8})=A_{1}\oplus E_{7}$
	$\mathfrak{g}(\alpha_{2})=A_{2}$

Table 2. Maximal regular subalgebras of maximal rank in simple Lie algebras (cf. [5, Table 12]). (In this table, the isomorphic subalgebras – e.g.

\mathfrak{g}(\alpha_{1})\simeq\mathfrak{g}(\alpha_{6})

E_{7}

– are not only isomorphic but conjugate by an inner automorphism.)

3.3. Dual pairs from maximal regular subalgebras of maximal rank

It turns out that maximal-rank maximal regular subalgebras lead to an important class of dual pairs.

Proposition 15 ([15, Proposition 5.15]).

Let $\mathfrak{g}$ be a simple Lie algebra. Let $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ be two semisimple subalgebras of $\mathfrak{g}$ such that $\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$ is a maximal regular subalgebra of maximal rank in $\mathfrak{g}$ . Then $(\mathfrak{g}_{1},\mathfrak{g}_{2})$ is a dual pair in $\mathfrak{g}$ .

Note that the adjectives “maximal-rank” and “maximal” are both required here, although “maximal-rank” was mistakenly omitted from the statement of [15, Proposition 5.15]. The original statement of [15, Proposition 5.15] would imply, for example, that there are dual pairs of the form $(A_{k},A_{n-1-k})$ in $A_{n}$ (for $k=1,\ldots,n-2$ ) coming from the maximal regular subalgebras $A_{k}\oplus A_{n-1-k}$ of rank $n-1$ in $A_{n}$ ⁴⁴4These maximal regular subalgebras (along with other maximal regular subalgebras of non-maximal rank) are mistakenly omitted from [5, Table 12].; however, it is not hard to check that these fail to be dual pairs. Similarly, removing the “maximal” adjective would imply, for example, that $(A_{1},2A_{3})$ is a dual pair in $E_{7}$ . However, from Table 2, it’s clear that $\mathfrak{z}_{E_{7}}(A_{1})\supseteq D_{6}\supsetneq 2A_{3}$ . (In fact, Proposition 15 implies that $\mathfrak{z}_{E_{7}}(A_{1})=D_{6}$ .) Therefore, by consulting Table 2, we see that Proposition 15 gives us the dual pairs shown in Table 3.

$\mathfrak{g}$	Dual Pair
$B_{n}$	$(D_{k},B_{n-k})$ : $k=2,3,\ldots,n-1$
	$(A_{1},A_{1}\oplus B_{n-2})$
$C_{n}$	$(C_{k},C_{n-k})$ : $k=1,2,\ldots,n-1$
$D_{n}$	$(D_{k},D_{n-k})$ : $k=2,3,\ldots,n-2$
	$(A_{1},A_{1}\oplus D_{n-2})$

$G_{2}$	$F_{4}$	$E_{6}$	$E_{7}$	$E_{8}$
$(A_{1},\widetilde{A_{1}})$	$(A_{1},C_{3})$	$(A_{1},A_{5})$	$(A_{1},D_{6})$	$(A_{4},A_{4})$
	$(A_{2},\widetilde{A_{2}})$	$(A_{2},2A_{2})$	$(A_{2},A_{5}^{{}^{\prime\prime}})$	$(A_{2},E_{6})$
				$(A_{1},E_{7})$

Table 3. Dual pairs coming from maximal regular subalgebras of maximal rank in simple Lie algebras.

Even though the adjective “maximal” is required for Proposition 15 to be correct, it can happen that non-maximal maximal-rank regular subalgebras can lead to dual pairs. For example, we will later see that $(B_{2},2A_{1})$ is a dual pair in $F_{4}$ , that $(D_{4},3A_{1})$ us a dual pair in $E_{7}$ , and that $(A_{5},A_{2}\oplus A_{1})$ , $(D_{6},2A_{1})$ , $(D_{5},A_{3})$ , $(D_{4},D_{4})$ , and $(D_{4}\oplus A_{1},3A_{1})$ are dual pairs in $E_{8}$ (see Tables 10, 12, and LABEL:table:E8). We will later see that these dual pairs (along with the ones in Table 3) are examples of non- $S$ -irreducible dual pairs.

Maximal-rank regular subalgebras will continue to play a crucial role in the classification of dual pairs throughout the remainder of the paper. In particular, it will turn out that a dual pair in a simple Lie algebra $\mathfrak{g}$ is either $S$ -irreducible in $\mathfrak{g}$ or $S$ -irreducible in a maximal-rank regular subalgebra of $\mathfrak{g}$ (see Theorem 28).

4. Admissible subalgebras

In this section, we introduce the notion of admissibility, which will play a crucial role in the classification of $S$ -irreducible dual pairs. To this end, let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$ , and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$ . Let $\mathcal{R}$ denote the root system of $(\mathfrak{g},\mathfrak{h})$ . Let

\Psi=\{\alpha_{1},\ldots,\alpha_{n}\}

be a basis (i.e. set of simple roots) for $\mathcal{R}$ .

Let $\theta$ denote some subset of $\Psi$ . Let $D$ denote the diagram obtained from the Dynkin diagram of $\Psi$ by circling the nodes corresponding to elements of $\Psi\setminus\theta$ . We will say that $D$ is the diagram associated with $(\Psi,\theta)$ . For each circled node $\gamma_{i}\in\Psi\setminus\theta$ , let $D_{i}$ denote the maximal connected subdiagram of $D$ whose unique circled node is $\gamma_{i}$ . Each $D_{i}$ is called an irreducible component of $D$ . For example, consider the $D_{9}$ Dynkin diagram with $\Psi\setminus\theta=\{\alpha_{2},\alpha_{5}\}$ :

This diagram has the following irreducible components:

Definition 16 ([15, Definition 3.1]).

The couple $(\Psi,\theta)$ is said to be admissible if the irreducible components $D_{i}$ of the diagram $D$ associated with $(\Psi,\theta)$ all appear in Figure 2. In this case $D$ will also be called admissible.

Remark 17.

The condition of admissibility corresponds to a certain associated prehomogeneous space being regular. See [15, Section 2] for more details.

Figure 2. A complete list of irreducible admissible diagrams [15, Table 1].

For a subset $\theta$ of $\Psi$ , set

\mathfrak{h}_{\theta}=\{H\in\mathfrak{h}\;|\;\alpha(H)=0\text{ for all }\alpha\in\theta\}.

Additionally, let $H_{\theta}\in\mathfrak{h}_{\theta}$ be uniquely defined by

\alpha(H_{\theta})=\left\{\begin{array}[]{ll}0&\text{if }\alpha\in\theta\\ 2&\text{if }\alpha\in\Psi\setminus\theta.\end{array}\right.

The following theorem shows that for each irreducible component $D_{i}$ of the diagram $D$ associated with $(\Psi,\theta)$ , the condition of appearing in Figure 2 is precisely the same as the existence of a certain $\mathfrak{sl}_{2}$ -triplet.

Theorem 18 ([15, Theorem 2.3]).

Suppose $\text{Card}(\Psi\setminus\theta)=1$ . Then the diagram $D$ associated with $(\Psi,\theta)$ appears in Figure 2 if and only if $H_{\theta}$ is a defining vector (i.e. if and only if there exist $X_{\theta},Y_{\theta}\in\mathfrak{g}$ such that $(X_{\theta},H_{\theta},Y_{\theta})$ is an $\mathfrak{sl}_{2}$ -triplet).

In this way, we see that if $(\Psi,\theta)$ is admissible, then we obtain a family of $\mathfrak{sl}_{2}$ -triplets, one for each irreducible component $D_{i}$ of the diagram $D$ associated with $(\Psi,\theta)$ .

Remark 19.

Note that for each irreducible component of a weighted diagram, the corresponding weighted diagram will have label 0 at every node except for the circled node, which will have label 2.

Theorem 20 ([15, Theorem 3.3]).

If the pair $(\Psi,\theta)$ is admissible, then the family of $\mathfrak{sl}_{2}$ -triplets $(Y_{i},H_{i},X_{i})$ generates a simple subalgebra $\widetilde{\mathfrak{g}_{\theta}}$ of $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}_{\theta}$ .

Example 21.

Let $\mathfrak{g}$ be the complex Lie algebra of type $E_{8}$ with Cartan subalgebra $\mathfrak{h}=\mathbb{C}^{8}$ . By [3, Plate VII], the root system corresponding to $(\mathfrak{g},\mathfrak{h})$ has the following simple roots:

	$\displaystyle\Psi=\Big{\{}\alpha_{1}$	$\displaystyle=\frac{1}{2}(\varepsilon_{1}+\varepsilon_{8})-\frac{1}{2}(\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}+\varepsilon_{7}),$
	$\displaystyle\alpha_{2}$	$\displaystyle=\varepsilon_{1}+\varepsilon_{2},\;\alpha_{3}=\varepsilon_{2}-\varepsilon_{1},\;\alpha_{4}=\varepsilon_{3}-\varepsilon_{2},\;\alpha_{5}=\varepsilon_{4}-\varepsilon_{3},$
	$\displaystyle\alpha_{6}$	$\displaystyle=\varepsilon_{5}-\varepsilon_{4},\;\alpha_{7}=\varepsilon_{6}-\varepsilon_{5},\;\alpha_{8}=\varepsilon_{7}-\varepsilon_{6}\Big{\}}.$

Let $\theta=\{\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{6},\alpha_{7},\alpha_{8}\}$ so that $\Psi\setminus\theta=\{\alpha_{1}\}$ . The diagram associated to $(\Psi,\theta)$ appears as 16) in Figure 2, and is consequently admissible. It is not hard to check that

\mathfrak{h}_{\theta}=\{H\in\mathfrak{h}\;|\;\alpha(H)=0\text{ for all }\alpha\in\theta\}=\left\{(0,0,0,0,0,0,0,x)\;|\;x\in\mathbb{C}\right\}.

Additionally, $H_{\theta}\in\mathfrak{h}_{\theta}$ is given by $H_{\theta}=(0,0,0,0,0,0,0,4)$ . By Theorem 20, $\widetilde{\mathfrak{g}_{\theta}}=\langle X_{\theta},H_{\theta},Y_{\theta}\rangle\simeq\mathfrak{sl}_{2}$ is a simple subalgebra of $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}_{\theta}$ (and defining vector $H_{\theta}$ ). Moreover, as explained in Remark 19, $\widetilde{\mathfrak{g}_{\theta}}$ has the following weighted diagram:

Therefore, [5, Table 20] gives that $\widetilde{\mathfrak{g}_{\theta}}=A_{1}^{8}$ .

5. $S$ -irreducible and admissible dual pairs

The goal of this section is to define the notion of an $S$ -irreducible dual pair, and to list all of the $S$ -irreducible dual pairs of the exceptional Lie algebras. To start, we define the notion of an $S$ -subalgebra, which was initially introduced in [5, No. 23].

Definition 22 ([15, Definition 5.9]).

Let $\mathfrak{g}$ be a semisimple Lie algebra. An $S$ -subalgebra of $\mathfrak{g}$ is a semisimple subalgebra that is not contained in a proper regular subalgebra of $\mathfrak{g}$ .

With this notion of $S$ -subalgebra, we can define a notion of irreducibility for dual pairs:

Definition 23 ([15, Definition 5.11]).

Let $(\mathfrak{a},\mathfrak{b})$ be a semisimple dual pair of a semisimple algebra $\mathfrak{g}$ . We say that the pair $(\mathfrak{a},\mathfrak{b})$ is $S$ -irreducible if the algebra $\mathfrak{a}\oplus\mathfrak{b}$ is an $S$ -subalgebra.

Remark 24.

In [8], Howe considers semisimple dual pairs in $\mathfrak{sp}_{2n}$ , and defines a notion of dual pair irreducibility in that setting. While the notion of dual pair irreducibility in Definition 23 a priori seems possibly different, it turns out that the two notions coincide in the setting of $\mathfrak{sp}_{2n}$ [15, Remark 5.13].

Figures 3 and 4 show all of the $S$ -subalgebras of the classical simple Lie algebras up to rank 6. Similarly, Figure 5 shows all of the $S$ -subalgebras of the exceptional Lie algebras, along with their inclusion relations. These figures, together with the following proposition, give us a method for easily identifying lots of $S$ -irreducible dual pairs:

Figure 3. Inclusion relations among the

S

-subalgebras of the classical simple Lie algebras up to rank 4 (cf. [13, Table XII]).

Figure 4. Inclusion relations among the

S

-subalgebras of the classical simple Lie algebras of ranks 5 and 6 (cf. [13, Table XII]).

Proposition 25 ([15, Proposition 5.19]).

Suppose $\mathfrak{g}$ is simple. Let $\mathfrak{a}$ and $\mathfrak{b}$ be semisimple subalgebras of $\mathfrak{g}$ such that $\mathfrak{a}\oplus\mathfrak{b}$ is a maximal $S$ -subalgebra. Then $(\mathfrak{a},\mathfrak{b})$ is an $S$ -irreducible dual pair in $\mathfrak{g}$ .

It is important to note that the converse of Proposition 25 does not hold (i.e. not all $S$ -irreducible dual pairs come from maximal $S$ -subalgebras).

Example 26.

As we saw in Example 21, $A_{1}^{8}$ is an admissible subalgebra of $E_{8}$ with $\Psi\setminus\theta=\{\alpha_{1}\}$ . By [10, Example 25, p. 146], we can deduce that $\mathfrak{z}_{E_{8}}(A_{1}^{8})=G_{2}^{1}\oplus G_{2}^{1}$ (cf. [15, Section 6.10]). From Figure 5, we see that $A_{1}^{8}\oplus G_{2}^{1}\oplus G_{2}^{1}$ is a (non-maxial) $S$ -subalgebra in $E_{8}$ . Additionally, Figure 5 and Proposition 25 imply that $(A_{1}^{8},G_{2}^{1})$ is a dual pair in $F_{4}$ and that $(G_{2}^{1},F_{4}^{1})$ is a dual pair in $E_{8}$ . Therefore,

\mathfrak{z}_{E_{8}}(G_{2}^{1}\oplus G_{2}^{1})=\mathfrak{z}_{F_{4}}(G_{2}^{1})=A_{1}^{8}.

In this way, we see that $(A_{1}^{8},G_{2}^{1}\oplus G_{2}^{1})$ is an $S$ -irreducible dual pair in $E_{8}$ .

As we will see in Theorem 27, this is where the notion of admissibility becomes useful.

Theorem 27 ([15, Theorem 8.1]).

Let $\mathfrak{g}$ be a simple Lie algebra and let $(\mathfrak{a},\mathfrak{b})$ be an $S$ -irreducible dual pair in $\mathfrak{g}$ . Then with the exception of the following cases, at least one of $\mathfrak{a}$ and $\mathfrak{b}$ is admissible:

(i)

$\mathfrak{g}$ is of type $D_{n}$ and $(\mathfrak{a},\mathfrak{b})$ is of type $(D_{k},D_{\ell})$ , where $n=2k\ell$ and $k,\ell\geq 2$ .
(ii)

$\mathfrak{g}$ is of type $D_{n}$ and $(\mathfrak{a},\mathfrak{b})$ is of type $(B_{p_{1}}\oplus B_{q_{1}},B_{p_{2}}\oplus B_{q_{2}})$ , where $(B_{p_{1}},B_{p_{2}})$ is an $S$ -irreducible dual pair in $B_{p}$ (with $2p+1=(2p_{1}+1)(2p_{2}+1)$ ), where $(B_{q_{1}},B_{q_{2}})$ is an $S$ -irreducible dual pair in $B_{q}$ (with $2q+1=(2q_{1}+1)(2q_{2}+1)$ ), and where $n=p+q+1$ .

Therefore, to generate a full list of $S$ -irreducible dual pairs in a given simple Lie algebra $\mathfrak{g}$ of any type other than $D_{n}$ , it suffices to consider all admissible diagrams associated with an appropriate $(\Psi,\theta)$ , and for each such diagram compute the corresponding subalgebra $\widetilde{\mathfrak{g}_{\theta}}$ . Finally, compute $\mathfrak{z}_{\mathfrak{g}}(\widetilde{\mathfrak{g}_{\theta}})$ and check whether $\widetilde{\mathfrak{g}_{\theta}}\oplus\mathfrak{z}_{\mathfrak{g}}(\widetilde{\mathfrak{g}_{\theta}})$ appears in Figure 5, 3, or 4 as an $S$ -subalgebra of $\mathfrak{g}$ . For $\mathfrak{g}$ of type $D_{n}$ , the $S$ -irreducibles are those obtained in this way in addition to those of the forms (i) and (ii) in Theorem 27 (see [15, Theorem 7.3.2] and [15, Proposition 7.3.3]). (Note that some admissible subalgebras are members of dual pairs that are not $S$ -irreducible. These dual pairs – along with the $S$ -irreducible dual pairs with an admissible factor – will be called admissible dual pairs).

Note that the “compute $\mathfrak{z}_{\mathfrak{g}}(\widetilde{\mathfrak{g}_{\theta}})$ ” step in this process is very nontrivial. Fortunately, the type of many of these centralizers can be deduced from [10]. Using this source, Rubenthaler carries out this process to classify the admissible dual pairs in the classical simple Lie algebras (see [15, Sections 6.3–6.7]) and in the exceptional Lie algebras (see [15, Sections 6.8–6.12]). These dual pairs will be included later on in the paper (see Section 6 for the classical simple Lie algebras and Section 9 for the exceptional Lie algebras).

6. Some of the dual pairs in the classical simple Lie algebras

Using what we have done so far, we can classify all of the dual pairs in simple Lie algebras coming from maximal regular subalgebras of maximal rank or having an admissible factor (which includes all $S$ -irreducible dual pairs for algebras of types other than $D_{n}$ ).

Recall that the dual pairs coming from maximal regular subalgebras of maximal rank are summarized in Table 3. In this section, we follow Rubenthaler in classifying the admissible dual pairs in the classical simple Lie algebras. Additionally, we explicitly write out these dual pairs (including index and “primes”) for the classical simple Lie algebras up to rank 6. These explicit dual pairs will be useful later on for our classification of dual pairs in the exceptional Lie algebras. Note that in the below tables and in what follows, whenever we encounter subalgebras of types $B_{1}$ , $C_{1}$ , $D_{1}$ , $C_{2}$ , or $D_{3}$ , we will instead write $A_{1}$ , $B_{2}$ , or $A_{3}$ so as to make these table entries more easily findable in Appendices A and B.

While we follow Rubenthaler’s classification approach in this section for notational consistency, it is worth noting that the dual pairs in the classical simple Lie algebras can also be understood more comprehensively using elementary methods. See [6] for more details.

6.1. Type $A_{n}$

Recall that the complex Lie algebra of type $A_{n}$ has no maximal-rank regular subalgebras. As we will soon see from Theorem 28, this means that all semisimple dual pairs in $A_{n}$ are $S$ -irreducible. By Theorem 27, any such dual pair has at least one factor that is an admissible subalgebra of $A_{n}$ . Moreover, it turns out that all admissible subalgebras of $A_{n}$ are $S$ -irreducible. Consequently, we obtain exactly the dual pairs outlined below (with the first factor having the following admissible diagram), where $n=(k+1)p-1$ , $k\geq 1$ , $n\geq 1$ , and $p\geq 2$ :

•
$S$ -irreducible:
- –
  
  $(A_{k},A_{p-1})$

The index of these subalgebras can be computed by hand or can be found in [13, Table XII] for $n\leq 6$ . Putting this all together, we get that the following is a complete list of dual pairs in $A_{n}$ (for $3\leq n\leq 6$ ).

Rank	Dual Pair	Rank	Dual Pair	Rank	Dual Pair	Rank	Dual Pair
3	$(A_{1}^{2},A_{1}^{2})$	4	N/A	5	$(A_{1}^{3},A_{2}^{2})$	6	N/A

Table 4. A complete list of semisimple dual pairs in

A_{n}

(for

3\leq n\leq 6

6.2. Type $B_{n}$

Recall that the complex Lie algebra of type $B_{n}$ has $(A_{1},A_{1}\oplus B_{n-2})$ and $(B_{n-k},D_{k})$ ( $k=2,3,\ldots,n$ ) as dual pairs arising from maximal regular subalgebras of maximal rank (see Table 3). By Theorem 27, any $S$ -irreducible dual pair in $B_{n}$ has at least one factor that is an admissible subalgebra of $B_{n}$ . Additionally, it turns out that some of the non- $S$ -irreducible dual pairs in $B_{n}$ have a factor that is an admissible subalgebra of $B_{n}$ . In particular, as we will see below, the $B_{n-k}$ factor of the dual pairs $(B_{n-k},D_{k})$ ( $k=2,3,\ldots,n$ ) is an admissible subalgebra of $B_{n}$ . In total, for type $B_{n}$ (with $n\geq 4$ ), the admissible dual pairs are outlined below (with the first factor having the following admissible diagram), where $n\geq kp$ , $(2k+1)p\leq 2n+1$ , $p\geq 1$ , and $p\neq 2$ :

•
$S$ -irreducible:
1. (a)
  
  $(B_{k},B_{\ell})$ : $p=2\ell+1$ ; $k,\ell\geq 1$ ; $2k\ell+k+\ell=n$ .
•
Non- $S$ -irreducible:
1. (i)
  
  $(B_{k},D_{\ell}\oplus B_{n-2k\ell-\ell})$ : $p=2\ell$ ; $k\geq 1$ ; $\ell\geq 2$ ; $2k\ell+\ell\leq n$ .
2. (ii)
  
  $(B_{k},B_{\ell}\oplus D_{n-2k\ell-k-\ell})$ : $p=2\ell+1$ ; $k\geq 1$ ; $\ell\geq 0$ ; $2k\ell+k+\ell+1<n$ .

The index of these subalgebras can be computed by hand or can be found in [13, Table XII] for $n\leq 6$ . Putting this all together, we get the following partial list of dual pairs in $B_{n}$ (for $2\leq n\leq 6$ ), including all $S$ -irreducible dual pairs and all dual pairs coming from maximal regular subalgebras of maximal rank.

Rank	Dual Pair	Rank	Dual Pair	Rank	Dual Pair	Rank	Dual Pair
2	$(A_{1},A_{1})$	4	$(A_{1},A_{1}\oplus B_{2}^{1^{\prime\prime}})$	5	$(A_{1},A_{1}\oplus B_{3})$	6	$(A_{1},A_{1}\oplus B_{4})$
3	$(\widetilde{A_{1}},2A_{1})$	4(ii)	$(A_{1}^{2^{\prime\prime}},A_{3}^{1^{\prime\prime}})$	5(ii)	$(A_{1}^{2^{\prime\prime}},D_{4})$	6(i)	$(A_{1}^{8},2A_{1}^{3^{\prime}})$
3	$(A_{1},A_{1}\oplus\widetilde{A_{1}})$	4(ii)	$(B_{2}^{1^{\prime\prime}},2A_{1})$	5(ii)	$(B_{2}^{1^{\prime\prime}},A_{3}^{1^{\prime\prime}})$	6(ii)	$(A_{1}^{2^{\prime\prime}},D_{5})$
		4(a)	$(A_{1}^{6},A_{1}^{6})$	5(ii)	$(B_{3},2A_{1})$	6(ii)	$(B_{2}^{1^{\prime\prime}},D_{4})$
						6(ii)	$(B_{3},A_{3}^{1^{\prime\prime}})$
						6(ii)	$(B_{4},2A_{1})$
						6(ii)	$(A_{1}^{6},A_{1}^{6}\oplus 2A_{1})$

Table 5. A partial list of dual pairs in

B_{n}

(for

2\leq n\leq 6

To find the remaining non- $S$ -irreducible dual pairs, one can carry out the process described in Section 7.

6.3. Type $C_{n}$

Recall that the complex Lie algebra of type $C_{n}$ has $(C_{k},C_{n-k})$ ( $k=1,2,\ldots,n-1$ ) as dual pairs arising from maximal regular subalgebras (see Table 3). By Theorem 27, any $S$ -irreducible dual pair in $C_{n}$ has at least one factor that is an admissible subalgebra of $C_{n}$ . Additionally, it turns out that some of the non- $S$ -irreducible dual pairs in $C_{n}$ have a factor that is an admissible subalgebra of $C_{n}$ . For type $C_{n}$ (with $n\geq 3$ ), the admissible dual pairs arise from two different admissible diagrams, as outlined below.

Type 1: We require that $p=2\ell$ and that $(2k+1)\ell\leq n$ .

•
$S$ -irreducible:
1. (a)
  
  $(B_{k},C_{\ell})$ : $k,\ell\geq 1$ ; $(2k+1)\ell=n$ .
•
Non- $S$ -irreducible:
1. (i)
  
  $(B_{k},C_{\ell}\oplus C_{n-2k\ell-\ell})$ : $k\geq 1$ ; $\ell\geq 1$ ; $(2k+1)\ell<n$ .

Type 2: We require that $n=(k+1)p$ , $k\geq 0$ , and $p\geq 1$ .

•
$S$ -irreducible:
1. (a)
  
  $(C_{k+1},B_{\ell})$ : $p=2\ell+1$ ; $k\geq 0$ ; $\ell\geq 1$ .
2. (b)
  
  $(C_{k+1},D_{\ell})$ : $p=2\ell$ ; $k\geq 0$ ; $\ell\geq 2$ .

The index of these subalgebras can be computed by hand or can be found in [13, Table XII] for $n\leq 6$ .

Putting this all together, we get the following partial list of dual pairs in $C_{n}$ (for $3\leq n\leq 6$ ), including all $S$ -irreducible dual pairs and all dual pairs coming from maximal regular subalgebras of maximal rank.

Rank	Dual Pair	Rank	Dual Pair	Rank	Dual Pair	Rank	Dual Pair
3	$(A_{1},B_{2})$	4	$(A_{1},C_{3})$	5	$(A_{1},C_{4})$	6	$(A_{1},C_{5})$
3(1a)	$(A_{1}^{8},A_{1}^{3})$	4	$(B_{2},B_{2})$	5	$(B_{2},C_{3})$	6	$(B_{2},C_{4})$
		4(1i)	$(A_{1}^{8},A_{1}^{3}\oplus A_{1})$	5(1i)	$(A_{1}^{8},A_{1}^{3}\oplus B_{2})$	6	$(C_{3},C_{3})$
		4(2b)	$(A_{1}^{4},2A_{1}^{4})$	5(1a)	$(B_{2}^{4},A_{1}^{5})$	6(1i)	$(A_{1}^{8},A_{1}^{3}\oplus C_{3})$
						6(1i)	$(B_{2}^{4},A_{1}^{5}\oplus A_{1})$
						6(1a)	$(A_{1}^{16},B_{2}^{3})$
						6(2b)	$(A_{1}^{6},A_{3}^{4})$

Table 6. A partial list of dual pairs in

C_{n}

(for

3\leq n\leq 6

To find the remaining non- $S$ -irreducible dual pairs, one can carry out the process described in Section 7.

6.4. Type $D_{n}$

Recall that the complex Lie algebra of type $D_{n}$ has $(A_{1},A_{1}\oplus D_{n-2})$ and $(D_{k},D_{n-k})$ ( $k=2,3,\ldots,n-2$ ) as dual pairs arising from maximal regular subalgebras (see Table 3). For type $D_{n}$ , Theorem 27 suggests that there are certain $S$ -irreducible dual pairs in $D_{n}$ that do not have admissible factors. In particular, $D_{n}$ has $S$ -irreducible dual pairs of type $(D_{k},D_{\ell})$ , where $n=2k\ell$ and $k,\ell\geq 2$ [15, Theorem 7.3.2]. Additionally, $D_{n}$ has $S$ -irreducible dual pairs of type $(B_{p_{1}}\oplus B_{q_{1}},B_{p_{2}}\oplus B_{q_{2}})$ , where $(B_{p_{1}},B_{p_{2}})$ is an $S$ -irreducible dual pair in $B_{p}$ (with $2p+1=(2p_{1}+1)(2p_{2}+1)$ ), where $(B_{q_{1}},B_{q_{2}})$ is an $S$ -irreducible dual pair in $B_{q}$ (with $2q+1=(2q_{1}+1)(2q_{2}+1)$ ), and where $n=p+q+1$ [15, Proposition 7.3.3]. However, note that neither of these types of dual pairs appear for $n\leq 6$ .

For all other $S$ -irreducible dual pairs in $D_{n}$ , Theorem 27 implies that at least one factor of the pair is an admissible subalgebra of $D_{n}$ . Additionally, it turns out that some of the non- $S$ -irreducible dual pairs in $D_{n}$ have a factor that is an admissible subalgebra of $D_{n}$ . For type $D_{n}$ (with $n\geq 4$ ), the admissible dual pairs arise from two different admissible diagrams, as outlined below.

Type 1: We require that $n\geq kp+2$ and that $(2k+1)p\leq 2n$ .

•
$S$ -irreducible:
1. (a)
  
  $(B_{k},D_{\ell})$ : $p=2\ell$ ; $k\geq 1$ ; $\ell\geq 2$ ; $(2k+1)\ell=n$ .
2. (b)
  
  $(B_{k},B_{\ell}\oplus B_{n-2k\ell-k-\ell-1})$ : $p=2\ell+1$ ; $k\geq 1$ ; $\ell\geq 0$ ; $2k\ell+k+\ell<n$ .
•
Non- $S$ -irreducible:
1. (i)
  
  $(B_{k},D_{\ell}\oplus D_{n-2k\ell-\ell})$ : $p=2\ell$ ; $k\geq 1$ ; $\ell\geq 3$ ; $(2k+1)\ell+1<n$ .

Type 2: We require that $n=(k+1)p$ and that $p=2\ell$ .

•
$S$ -irreducible:
1. (a)
  
  $(C_{k+1},C_{\ell})$ : $k\geq 0$ ; $\ell\geq 1$ .

The index of these subalgebras can be computed by hand or can be found in [13, Table XII] for $n\leq 6$ .

Putting this all together, we get the following partial list of dual pairs in $D_{n}$ (for $4\leq n\leq 6$ ), including all $S$ -irreducible dual pairs and all dual pairs coming from maximal regular subalgebras of maximal rank.

Rank	Dual Pair	Rank	Dual Pair	Rank	Dual Pair
4	$(A_{1},3A_{1})$	5	$(A_{1},A_{1}\oplus A_{3}^{{}^{\prime\prime}})$	6	$(A_{1},A_{1}\oplus D_{4})$
4	$(2A_{1},2A_{1})$	5	$(2A_{1},A_{3}^{{}^{\prime\prime}})$	6	$(2A_{1},D_{4})$
4(1b)	$(A_{1}^{2^{\prime}},B_{2}^{1^{\prime}})$	5(1b)	$(A_{1}^{2^{\prime\prime}},B_{3}^{1})$	6	$(A_{3}^{{}^{\prime\prime}},A_{3}^{{}^{\prime\prime}})$
4(1b)	$(A_{1}^{2^{\prime\prime}},B_{2}^{1^{\prime\prime}})$	5(1b)	$(B_{2}^{1^{\prime\prime}},B_{2}^{1^{\prime\prime}})$	6(1a)	$(A_{1}^{8},A_{1}^{3^{\prime}}\oplus A_{1}^{3^{\prime\prime}})$
4(1b)	$(A_{1}^{2^{\prime\prime\prime}},B_{2}^{1^{\prime\prime\prime}})$	5(1b)	$(A_{1}^{6},A_{1}^{6})$	6(1b)	$(A_{1}^{2^{\prime\prime}},B_{4}^{1})$
				6(1b)	$(B_{2}^{1^{\prime\prime}},B_{3}^{1})$
				6(1b)	$(A_{1}^{6},A_{1}^{6}\oplus A_{1}^{2^{\prime\prime}})$
				6(2a)	$(A_{1}^{3^{\prime}},C_{3}^{1^{\prime\prime}})$
				6(2a)	$(A_{1}^{3^{\prime\prime}},C_{3}^{1^{\prime}})$

Table 7. A partial list of dual pairs in

D_{n}

(for

4\leq n\leq 6

To find the remaining non- $S$ -irreducible dual pairs, one can carry out the process described in Section 7.

7. Non- $S$ -irreducible dual pairs

So far, we have established how to classify dual pairs coming from maximal regular subalgebras of maximal rank (see Table 3) and how to classify all dual pairs with an admissible factor (see Section 5). In this section, we discuss how to classify the remaining non- $S$ -irreducible dual pairs.

Theorem 28 ([15, Theorem 5.12]).

Let $(\mathfrak{a},\mathfrak{b})$ be a semisimple non- $S$ -irreducible dual pair in a semisimple $\mathfrak{g}$ . Then there exists a maximal-rank regular semisimple subalgebra $\mathfrak{u}$ in $\mathfrak{g}$ such that $\mathfrak{a}\oplus\mathfrak{b}\subset\mathfrak{u}\subset\mathfrak{g}$ and such that $\mathfrak{a}\oplus\mathfrak{b}$ is an $S$ -subalgebra of $\mathfrak{u}$ .

Note that the subalgebra $\mathfrak{u}$ in this theorem is maximal-rank and regular but need not be maximal regular. This theorem shows that to find all of the non- $S$ -irreducible dual pairs in a simple Lie algebra $\mathfrak{g}$ , one can carry out the following process:

(1)

Write down all of the maximal-rank regular subalgebras of $\mathfrak{g}$ (found in Table 1).
(2)

For each maximal-rank regular subalgebra $\mathfrak{u}$ of $\mathfrak{g}$ , find all $S$ -subalgebras $\mathfrak{a}\oplus\mathfrak{b}$ of $\mathfrak{u}$ .
(3)

For each pair $(\mathfrak{a},\mathfrak{b})$ from step 2, determine whether $(\mathfrak{a},\mathfrak{b})$ is a dual pair in $\mathfrak{g}$ .

Using Table 3, step 1 is straightforward. For step 2, the $S$ -subalgebras of $\mathfrak{u}$ can be found in Figure 3 or 4 if $\mathfrak{u}$ is a classical simple Lie algebra of rank at most 6 and can be found in Figure 5 if $\mathfrak{u}$ is an exceptional Lie algebra. This information (together with the following lemma) can be used to find the $S$ -subalgebras of the semisimple Lie algebras $\mathfrak{u}$ we will consider.

Lemma 29.

Let $\mathfrak{u}_{1}$ and $\mathfrak{u}_{2}$ be complex semisimple Lie algebras.

(a)

Let $s_{1}$ (resp. $s_{2}$ ) be an $S$ -subalgebra of $\mathfrak{u}_{1}$ (resp. $\mathfrak{u}_{2}$ ). Then $s_{1}\oplus s_{2}$ is an $S$ -subalgebra of $\mathfrak{u}_{1}\oplus\mathfrak{u}_{2}$ .
(b)

Let $s_{1}\oplus s_{2}$ be an $S$ -subalgebra of $\mathfrak{u}_{1}\oplus\mathfrak{u}_{2}$ . Then $s_{1}$ is an $S$ -subalgebra of $\mathfrak{u}_{1}$ and $s_{2}$ is an $S$ -subalgebra of $\mathfrak{u}_{2}$ .

Proof of (a).

Suppose, for the sake of contradiction, that $s_{1}\oplus s_{2}$ is contained in some proper regular subalgebra $r_{1}\oplus r_{2}$ of $\mathfrak{u}_{1}\oplus\mathfrak{u}_{2}$ . Then there exists a Cartan subalgebra $\mathfrak{h}_{1}\oplus\mathfrak{h}_{2}$ of $\mathfrak{u}_{1}\oplus\mathfrak{u}_{2}$ such that

[\mathfrak{h}_{1}\oplus\mathfrak{h}_{2},r_{1}\oplus r_{2}]=[\mathfrak{h}_{1},r_{1}]\oplus[\mathfrak{h}_{2},r_{2}]\subset r_{1}\oplus r_{2}.

It is not hard to see that $\mathfrak{h}_{1}$ and $\mathfrak{h}_{2}$ are Cartan subalgebras of $\mathfrak{u}_{1}$ and $\mathfrak{u}_{2}$ , respectively, and hence that $r_{1}$ and $r_{2}$ are regular subalgebras of $\mathfrak{u}_{1}$ and $\mathfrak{u}_{2}$ containing $s_{1}$ and $s_{2}$ , respectively. Moreover, since $r_{1}\oplus r_{2}$ is proper in $\mathfrak{u}_{1}\oplus\mathfrak{u}_{2}$ , at least one of $r_{1}$ and $r_{2}$ has to be proper. This then gives a contradiction, proving that $s_{1}\oplus s_{2}$ is an $S$ -subalgebra of $\mathfrak{u}_{1}\oplus\mathfrak{u}_{2}$ . ∎

Proof of (b).

Suppose, for the sake of contradiction, that $s_{1}$ is contained in a proper regular subalgebra $r_{1}$ of $\mathfrak{u}_{1}$ . Then $r_{1}\oplus\mathfrak{u}_{2}$ is clearly a proper regular subalgebra of $\mathfrak{u}_{1}\oplus\mathfrak{u}_{2}$ containing $s_{1}\oplus s_{2}$ , a contradiction. Repeating this argument with the roles of $\mathfrak{u}_{1}$ and $\mathfrak{u}_{2}$ reversed, we again obtain a contradiction. ∎

For step 3, we first need to view the $S$ -subalgebras of $\mathfrak{u}$ as subalgebras of $\mathfrak{g}$ . For a simple $S$ -subalgebra $\mathfrak{a}$ of a simple factor of $\mathfrak{u}$ , we can apply the formula $i_{\mathfrak{a}\hookrightarrow\mathfrak{g}}=i_{\mathfrak{a}\hookrightarrow\mathfrak{u}}\cdot i_{\mathfrak{u}\hookrightarrow\mathfrak{g}}$ to find the index of $\mathfrak{a}$ in $\mathfrak{g}$ . Note that since the simple factors of $\mathfrak{u}$ will almost always have index 1 in $\mathfrak{g}$ (see Remark 11), we will usually have that $i_{\mathfrak{a}\hookrightarrow\mathfrak{u}}=i_{\mathfrak{a}\hookrightarrow\mathfrak{g}}$ . While the value of $i_{\mathfrak{a}\hookrightarrow\mathfrak{g}}$ is easy to find, it can often be more challenging to figure out the conjugacy class of $\mathfrak{a}$ in situations where the conjugacy class is not uniquely determined by the type and index of $\mathfrak{a}$ . This challenge will be addressed in several of the examples in the following section. Finally, the remainder of step 3 (i.e. determining whether a candidate dual pair is actually a dual pair) is even more invovled. The next section is primarily dedicated to discussing different methods for confirming and eliminating these candidate dual pairs for the exceptional Lie algebras.

8. Eliminating and confirming candidate dual pairs

8.1. Straightforward eliminations

Perhaps the most straightforward approach for ruling out a candidate dual pair $(\mathfrak{a},\mathfrak{b})$ in $\mathfrak{g}$ is to show that $\mathfrak{z}_{\mathfrak{g}}(\mathfrak{a})\supsetneq\mathfrak{b}$ using information about subalgebras and dual pairs that have already been confirmed. For example, whenever $\mathfrak{a}\oplus\mathfrak{a}^{\prime}$ is a subalgebra of $\mathfrak{g}$ with $\mathfrak{a}^{\prime}\supsetneq\mathfrak{b}$ , this gives that

\mathfrak{z}_{\mathfrak{g}}(\mathfrak{a})\supseteq\mathfrak{a}^{\prime}\supsetneq\mathfrak{b},

so we can conclude that $(\mathfrak{a},\mathfrak{b})$ is not a dual pair in $\mathfrak{g}$ .

Similarly, suppose that $(\mathfrak{a},\mathfrak{b})$ is a dual pair in $\mathfrak{g}$ and that $\mathfrak{b}_{1}\oplus\mathfrak{b}_{2}$ is a subalgebra of $\mathfrak{b}$ . This situation often leads to a candidate dual pair in $\mathfrak{g}$ of the form $(\mathfrak{a}\oplus\mathfrak{b}_{1},\mathfrak{b}_{2})$ (e.g. when $\mathfrak{a}\oplus\mathfrak{b}$ is regular of maximal rank in $\mathfrak{g}$ and $\mathfrak{b}_{1}\oplus\mathfrak{b}_{2}$ is an $S$ -subalgebra of $\mathfrak{b}$ ). Since $(\mathfrak{a},\mathfrak{b})$ is a dual pair in $\mathfrak{g}$ , we have that

\mathfrak{z}_{\mathfrak{g}}(\mathfrak{a}\oplus\mathfrak{b}_{1})=\mathfrak{z}_{\mathfrak{b}}(\mathfrak{b}_{1})

and that

\mathfrak{z}_{\mathfrak{g}}(\mathfrak{b}_{2})\supseteq\mathfrak{z}_{\mathfrak{a}\oplus\mathfrak{b}}(\mathfrak{b}_{2})=\mathfrak{a}\oplus\mathfrak{z}_{\mathfrak{b}}(\mathfrak{b}_{2}).

In this way, we see that for $(\mathfrak{a}\oplus\mathfrak{b}_{1},\mathfrak{b}_{2})$ to be a dual pair in $\mathfrak{g}$ , it is necessary that $(\mathfrak{b}_{1},\mathfrak{b}_{2})$ is a dual pair in $\mathfrak{b}$ . (Note, however, that this condition is not enough to guarantee that $(\mathfrak{a}\oplus\mathfrak{b}_{1},\mathfrak{b}_{2})$ is a dual pair in $\mathfrak{g}$ .) This idea is demonstrated in the following example:

Example 30 ( $(A_{1}^{2}\oplus 3A_{1},B_{2}^{1})$ is not a dual pair in $E_{7}$ ).

Since $(B_{2}^{1^{\prime\prime}},B_{3}^{1})$ is an $S$ -irreducible dual pair in $D_{6}$ , we have that $B_{2}^{1}\oplus B_{3}^{1}\oplus A_{1}$ is an $S$ -subalgebra of $E_{7}$ , and hence that

\mathfrak{z}_{E_{7}}(B_{2}^{1})\supseteq B_{3}^{1}\oplus A_{1}\supsetneq A_{1}^{2}\oplus 3A_{1}.

Therefore, $(A_{1}^{2}\oplus 3A_{1},B_{2}^{1})$ is not a dual pair in $E_{7}$ .

8.2. Eliminating and confirming using Dynkin’s and Carter’s tables

To augment our strategy of using information about known dual pairs to make straightforward eliminations of candidate dual pairs, we can reference [5, Table 25] and the tables in [4, pp. 401–405]. The tables in [4, pp. 401–405] indicate the type of the centralizer of each 3-dimensional subalgebra of the exceptional Lie algebras, and [5, Table 25] lists all of the conjugacy classes of subalgebras of the exceptional Lie algebras.

Example 31 ( $(A_{3},\widetilde{A_{1}})$ is a dual pair in $F_{4}$ ).

Recall that $A_{3}\oplus\widetilde{A_{1}}$ is a maximal-rank regular subalgebra in $F_{4}$ , but not a maximal regular subalgebra. Therefore, $(A_{3},\widetilde{A_{1}})$ is a candidate dual pair in $F_{4}$ , but Proposition 15 does not apply. Fortunately, by [4, p. 401], we have that the centralizer of $\widetilde{A_{1}}$ in $F_{4}$ has type $A_{3}$ . Moreover, by [5, Table 25], there is a unique (up to conjugation) subalgebra of type $A_{3}$ in $F_{4}$ , and it has index 1. On the other hand, the centralizer of $A_{3}$ in $F_{4}$ clearly contains $\widetilde{A_{1}}$ and has rank at most 1, so we can conclude that $\mathfrak{z}_{F_{4}}(A_{3})=\widetilde{A_{1}}$ . It follows that $(A_{3},\widetilde{A_{1}})$ is a dual pair in $F_{4}$ .

Example 32 ( $(\widetilde{A_{1}},B_{2}^{1})$ and $(\widetilde{A_{1}},2\widetilde{A_{1}})$ are not dual pairs in $F_{4}$ ).

Recall that $(A_{1}^{2^{\prime\prime}},B_{2}^{1^{\prime\prime}})$ is an $S$ -irreducible dual pair in $D_{4}$ (see Table 7). Since $D_{4}$ is a maximal-rank regular subalgebra of index 1 in $F_{4}$ , $(A_{1}^{2},B_{2}^{1})$ is a candidate dual pair in $F_{4}$ . However, by [5, Table 20], there is a unique (up to conjugation) subalgebra of type $A_{1}$ and index 2 in $F_{4}$ (namely, $\widetilde{A_{1}}$ ). Since we already know that the centralizer of $\widetilde{A_{1}}$ in $F_{4}$ is $A_{3}$ (by [4, p. 401] or Example 31), it follows that $(A_{1}^{2},B_{2}^{1})=(\widetilde{A_{1}},B_{2}^{1})$ is not a dual pair in $F_{4}$ . Similarly, $(A_{1}^{2},2A_{1}^{2})=(\widetilde{A_{1}},2\widetilde{A_{1}})$ is not a dual pair in $F_{4}$ .

8.3. Eliminating and confirming using dimension of the centralizer

Let $\mathfrak{g}$ be a complex simple Lie algebra, and let $\widetilde{\mathfrak{g}}$ be a semisimple subalgebra of $\mathfrak{g}$ . The representation which is induced on $\widetilde{\mathfrak{g}}$ by the adjoint representation of $\mathfrak{g}$ splits into two components: one component acts on $\widetilde{\mathfrak{g}}$ and can be thought of as the adjoint representation of $\widetilde{\mathfrak{g}}$ ; the complementary component, called the characteristic representation of $\widetilde{\mathfrak{g}}$ , is denoted by $\chi_{\widetilde{\mathfrak{g}}}$ [5, No. 6].

Lemma 33.

Let $M$ denote the number of copies of the trivial representation in $\chi_{\widetilde{\mathfrak{g}}}$ . Then

M=\dim\mathfrak{z}_{\mathfrak{g}}(\widetilde{\mathfrak{g}}).

Proof.

Let $G$ be the simply connected Lie group associated to $\mathfrak{g}$ , and let $\widetilde{G}\subset G$ be the simply connected Lie group associated to $\widetilde{\mathfrak{g}}$ . We have that

\operatorname{Res}^{G}_{\widetilde{G}}(\text{Ad}_{G})=\text{Ad}_{\widetilde{G}}\oplus\chi_{\widetilde{G}},

where $\text{Ad}_{G}:G\rightarrow\text{Aut}(\mathfrak{g})$ is the adjoint representation of $G$ , where $\text{Ad}_{\widetilde{G}}:\widetilde{G}\rightarrow\text{Aut}(\widetilde{\mathfrak{g}})$ is the adjoint representation of $\widetilde{G}$ , and where $\chi_{\widetilde{G}}:\widetilde{G}\rightarrow\text{Aut}(\mathfrak{g}/\widetilde{\mathfrak{g}})$ is the resulting complementary component. Noting that $\chi_{\widetilde{\mathfrak{g}}}=d\chi_{\widetilde{G}}$ , we see that the number of copies of the trivial representation in $\chi_{\widetilde{\mathfrak{g}}}$ equals the number of copies of the trivial representation in $\chi_{\widetilde{G}}$ .

The set of copies of the trivial representation in $\chi_{\widetilde{G}}$ correspond to a set of linearly independent vectors, each of which is fixed by all of $\chi_{\widetilde{G}}(\widetilde{G})$ . Indeed, any vector that is fixed by all of $\chi_{\widetilde{G}}(\widetilde{G})$ spans a one-dimensional subrepresentation, which is necessarily the trivial representation. Moreover, we note that each such fixed vector corresponds to an element of $\mathfrak{z}_{\mathfrak{g}}(\widetilde{\mathfrak{g}})$ . Indeed, for $Y\in\mathfrak{g}/\widetilde{\mathfrak{g}}$ , we have that

e^{X}Ye^{-X}=Y\;\text{ for all }X\in\widetilde{\mathfrak{g}}\hskip 14.22636pt\iff\hskip 14.22636pt[X,Y]=0\;\text{ for all }X\in\widetilde{\mathfrak{g}}.

Finally, since $\widetilde{\mathfrak{g}}$ is semisimple, we have that $\widetilde{\mathfrak{g}}\cap\mathfrak{z}_{\mathfrak{g}}(\widetilde{\mathfrak{g}})=\varnothing$ , so these elements of $\mathfrak{g}/\widetilde{\mathfrak{g}}$ span $\mathfrak{z}_{\mathfrak{g}}(\widetilde{\mathfrak{g}})$ . It follows that $M=\dim\mathfrak{z}_{\mathfrak{g}}(\widetilde{\mathfrak{g}})$ , as desired. ∎

The decomposition of $\chi_{\widetilde{\mathfrak{g}}}$ into irreducibles is included in [5, Table 25] for subalgebras of the exceptional Lie algebras. The number $M$ in the previous lemma can be read off of this table as the coefficient of the trivial representation, which Dynkin denotes by $N$ . This gives us an easy way to look up the dimension of $\mathfrak{z}_{\mathfrak{g}}(\widetilde{\mathfrak{g}})$ for such subalgebras, which in some cases allows us to confirm or eliminate candidate dual pairs.

Example 34 ( $(A_{2}^{3},A_{2}^{3})$ is not a dual pair in $E_{8}$ ).

By Subsection 6.1, there is an $S$ -irreducible dual pair in $A_{8}$ where both factors have type $A_{2}$ and index 3. By [5, Table 25], $E_{8}$ has two non-conjugate subalgebras of type $A_{2}$ and index 3 (i.e. $A_{2}^{3^{\prime}}$ and $A_{2}^{3^{\prime\prime}}$ ). However, by the same table, $\mathfrak{z}_{E_{8}}(A_{2}^{3^{\prime}})$ has dimension 28 and $\mathfrak{z}_{E_{8}}(A_{2}^{3^{\prime\prime}})$ has dimension 14. Since $\dim A_{2}=8$ , we therefore see that neither of these possibilities leads to a dual pair in $E_{8}$ .

Example 35 ( $(B_{3}^{1},B_{4}^{1})$ is a dual pair in $E_{8}$ ).

By Subsection 6.4, there is an $S$ -irreducible dual pair in $D_{8}$ where one factor has type $B_{3}$ and index 1 and the other factor has type $B_{4}$ and index 1. By [5, Table 25], these types and index values uniquely specify subalgebras of $E_{8}$ (up to conjugation). By the same table, $\dim\mathfrak{z}_{E_{8}}(B_{3}^{1})=36=\dim B_{4}$ and $\dim\mathfrak{z}_{E_{8}}(B_{4}^{1})=21=\dim B_{3}$ . From this, it’s clear that $(B_{3}^{1},B_{4}^{1})$ is a dual pair in $E_{8}$ .

Example 36 ( $(A_{1}^{3},A_{2}^{2^{\prime\prime}}\oplus A_{1})$ is a dual pair in $E_{6}$ ).

By Table 4, $(A_{1}^{3},A_{2}^{2})$ is an $S$ -irreducible dual pair in $A_{5}$ . Since $A_{5}\oplus A_{1}$ is a maximal regular subalgebra of maximal rank in $E_{6}$ , $(A_{5},A_{1})$ is a dual pair in $E_{6}$ and there is a candidate dual pair in $E_{6}$ with the following types and index values: $(A_{1}^{3},A_{2}^{2}\oplus A_{1})$ . By [5, Table 25], there is a unique (up to conjugation) subalgebra of type $A_{1}$ and index 3 in $E_{6}$ , as well as a unique (up to conjugation) subalgebra of type $A_{1}$ and index 1 in $E_{6}$ ; however, there are two conjugacy classes of subalgebras of type $A_{2}$ and index 2. Fortunately, with Lemma 33, [5, Table 25], and some further investigation, we can determine which conjugacy class we are dealing with here.

To this end, consider the extended Dynkin diagram for $E_{6}$ :

Crossing out the node corresponding to $\alpha_{2}$ , we obtain the maximal regular subalgebra of type $A_{5}\oplus A_{1}$ in $E_{6}$ . To understand $A_{1}^{3}$ and this $A_{2}^{2}$ as subalgebras of $E_{6}$ , we can consider the following diagram (where the admissible diagram corresponding to $A_{1}^{3}$ in $A_{5}$ is indicated in blue):

Realizing the $A_{5}$ root system as in [3, Plate I], it is not hard to show that the defining vector for $A_{1}^{3}$ is $(1,1,1,-1,-1,-1)$ . With this, we see that the centralizer of $A_{1}^{3}$ in $A_{5}$ has type $A_{2}$ and is diagonally embedded in the subalgebra of type $A_{2}\oplus A_{2}$ coming from $\alpha_{1}$ , $\alpha_{2}$ , $\alpha_{5}$ , and $\alpha_{6}$ in $E_{6}$ . In this way, we see that the centralizer of this $A_{2}^{2}$ in $E_{6}$ should contain $A_{1}^{3}$ in addition to a subalgebra of type $A_{2}$ coming from $\alpha_{2}$ and $\widetilde{\alpha}$ (since the nodes corresponding to $\alpha_{2}$ and $\widetilde{\alpha}$ are not adjacent to $\alpha_{1}$ , $\alpha_{2}$ , $\alpha_{5}$ , or $\alpha_{6}$ ). Therefore, the dimension of the centralizer of this $A_{2}^{2}$ is at least 11. Consulting [5, Table 25], we see that this subalgebra must therefore be $A_{2}^{2^{\prime\prime}}$ .

Finally, [4, p. 402] gives that $\mathfrak{z}_{E_{6}}(A_{1}^{3})$ has type $A_{2}\oplus A_{1}$ , and

\mathfrak{z}_{E_{6}}(A_{2}^{2^{\prime\prime}}\oplus A_{1})=\mathfrak{z}_{A_{5}}(A_{2}^{2})=A_{1}^{3},

so we can conclude that $(A_{1}^{3},A_{2}^{2^{\prime\prime}}\oplus A_{1})$ is a dual pair in $E_{6}$ .

8.4. Eliminating and confirming based on fixed vectors of $\chi_{\widetilde{\mathfrak{g}}}$

While looking up the dimension of centralizers in [5, Table 25] (using Lemma 33) helps us confirm and eliminate many candidate dual pairs, there are a few cases in which we need to compute $\chi_{\widetilde{\mathfrak{g}}}$ more explicitly. For example, consider the candidate dual pair in $E_{7}$ with types and index values $(A_{2}^{2},A_{1}^{3}\oplus A_{2})$ (to be explored in more detail in Example 38). By [5, Table 25], there are two conjugacy classes of subalgebras of type $A_{2}$ and index 2 in $E_{7}$ . Moreover, while we know that the centralizer of the $A_{2}^{2}$ in question contains $A_{1}^{3}\oplus A_{2}$ , Lemma 33 and [5, Table 25] show that the centralizers of both $A_{2}^{2^{\prime}}$ and $A_{2}^{2^{\prime\prime}}$ have dimension at least 11 (so we cannot conclude whether we’re working with $A_{2}^{2^{\prime}}$ or $A_{2}^{2^{\prime\prime}}$ based on this information).

Remark 37.

In situations like the one just described, we can calculate $\chi_{\widetilde{\mathfrak{g}}}$ by hand to check for copies of the trivial representation in $\chi_{\widetilde{\mathfrak{g}}}$ and to determine the conjugacy class of $\widetilde{\mathfrak{g}}$ .

Example 38 ( $(A_{2}^{2^{\prime}},A_{1}^{3^{\prime\prime}}\oplus A_{2})$ is a dual pair in $E_{7}$ ).

Consider once again the candidate dual pair in $E_{7}$ with types and index values $(A_{2}^{2},A_{1}^{3}\oplus A_{2})$ . To determine the conjugacy class of $A_{1}^{3}$ , we can look at the embeddings $A_{1}^{3}\subset A_{5}^{{}^{\prime\prime}}\subset E_{7}$ . To this end, consider the extended Dynkin diagram for $E_{7}$ :

Crossing out the node corresponding to $\alpha_{3}$ , we obtain the maximal regular subalgebra of type $A_{2}\oplus A_{5}^{{}^{\prime\prime}}$ in $E_{7}$ . To understand this index-3 subalgebra of type $A_{1}$ , we can consider the following diagram (where the admissible diagram corresponding to $A_{1}^{3}$ in $A_{5}^{{}^{\prime\prime}}$ is indicated in blue):

We can realize the $A_{5}$ root system as in [3, Plate I]. With this, it is not hard to show that this $A_{1}^{3}$ has defining vector $(1,1,1,-1,-1,-1)$ . Putting this vector in dominant form, we see that this subalgebra has weighted diagram

in $E_{7}$ , meaning the subalgebra in question is $A_{1}^{3^{\prime\prime}}$ [5, Table 19].

For $A_{2}^{2}$ , it is a bit more difficult to figure out which conjugacy class we’re working with. However, the information about how this $A_{2}^{2}$ sits inside of $E_{7}$ allows us to determine the number of copies of the trivial representation in $\chi_{A_{2}^{2}}$ without knowing beforehand which conjugacy class this $A_{2}^{2}$ belongs to. To this end, let $\mathfrak{s}=A_{2}^{2}\oplus A_{1}^{3^{\prime\prime}}\oplus A_{2}$ , and let

S=SU(3,\mathbb{C})\times SU(2,\mathbb{C})\times SU(3,\mathbb{C}).

Note that by the same reasoning as in Lemma 33, $\operatorname{Res}^{\mathfrak{s}}_{A_{2}^{2}}(\chi_{\mathfrak{s}})$ has no copies of the trivial representation if and only if $\operatorname{Res}^{S}_{SU(3,\mathbb{C})}(\chi_{S})$ has no fixed vectors if and only if $\mathfrak{z}_{E_{7}}(A_{2}^{2})=A_{1}^{3^{\prime\prime}}\oplus A_{2}$ . We have that $\chi_{S}$ acts on the space $E_{7}/(A_{2}^{2}\oplus A_{1}^{3^{\prime\prime}}\oplus A_{2})$ , and that

E_{7}/(A_{2}^{2}\oplus A_{1}^{3^{\prime\prime}}\oplus A_{2})\simeq[E_{7}/(A_{5}^{{}^{\prime\prime}}\oplus A_{2})]\oplus[A_{5}^{{}^{\prime\prime}}/(A_{2}^{2}\oplus A_{1}^{3})].

Now, since $(A_{2}^{2},A_{1}^{3})$ is a dual pair in $A_{5}$ , $\operatorname{Res}^{S}_{SU(3,\mathbb{C})}(\chi_{S})$ will not have any fixed vectors coming from $A_{5}^{{}^{\prime\prime}}/(A_{2}^{2}\oplus A_{1}^{3})$ . The $E_{7}/(A_{5}^{{}^{\prime\prime}}\oplus A_{2})$ -factor of $\chi_{S}$ decomposes in terms of fundamental representations of $SU(3,\mathbb{C})$ and $SU(2,\mathbb{C})$ as follows:

[\wedge^{2}(\mathbb{C}^{3}\otimes\mathbb{C}^{2})\otimes\mathbb{C}^{3}]\oplus[(\wedge^{2}(\mathbb{C}^{3}\otimes\mathbb{C}^{2}))^{*}\otimes(\mathbb{C}^{3})^{*}].

Letting $\{(1,0,0),(0,1,0),(0,0,1)\}$ and $\{(1,0),(0,1)\}$ be the standard weight bases for $\mathbb{C}^{3}$ and $\mathbb{C}^{2}$ , respectively, we get the following weight basis for $\mathbb{C}^{3}\otimes\mathbb{C}^{2}$ :

	$\displaystyle\{$	$\displaystyle(1,0,0,1,0),\,(0,1,0,1,0),\,(0,0,1,1,0),$
		$\displaystyle(1,0,0,0,1),\,(0,1,0,0,1),\,(0,0,1,0,1)\}.$

A weight basis for $\wedge^{2}(\mathbb{C}^{3}\otimes\mathbb{C}^{2})$ is then obtained by taking sums of any two distinct elements from the $\mathbb{C}^{3}\otimes\mathbb{C}^{2}$ weight basis. We get $(2,0,0,1,1)$ and $(1,1,0,2,0)$ as highest weights, which have weight spaces of dimensions 6 and 9, respectively (by the Weyl dimension formula). In particular, when decomposing the $E_{7}/(A_{5}^{{}^{\prime\prime}}\oplus A_{2})$ -factor of $\chi_{S}$ into irreducibles, we do not get any copies of the trivial representation of the $SU(3,\mathbb{C})$ corresponding to $A_{2}^{2}$ . It follows that $\mathfrak{z}_{E_{7}}(A_{2}^{2})=A_{1}^{3^{\prime\prime}}\oplus A_{2}$ . Finally, by [5, Table 25] and Lemma 33, we see that this index-2 subalgebra of type $A_{2}$ is $A_{2}^{2^{\prime}}$ .

9. All of the semisimple dual pairs in the exceptional Lie algebras

9.1. $S$ -irreducible dual pairs in the exceptional Lie algebras

As described in Section 5, one can generate a complete list of the $S$ -irreducible dual pairs in an exceptional Lie algebra $\mathfrak{g}$ by considering all admissible diagrams associated with an appropriate $(\Psi,\theta)$ , computing the subalgebra $\widetilde{\mathfrak{g}_{\theta}}$ corresponding to $(\Psi,\theta)$ , computing $\mathfrak{z}_{\mathfrak{g}}(\widetilde{\mathfrak{g}_{\theta}})$ and checking whether $\widetilde{\mathfrak{g}_{\theta}}\oplus\mathfrak{z}_{\mathfrak{g}}(\widetilde{\mathfrak{g}_{\theta}})$ appears in Figure 5 as an $S$ -subalgebra of $\mathfrak{g}$ . Rubenthaler carries out this process in [15, Sections 6.8–6.12], and the results are summarized in Table 8.

Figure 5. Inclusion relations among the

S

-subalgebras of the exceptional Lie algebras [5, Table 39].

$\mathfrak{g}$	Dual Pair $(\widetilde{\mathfrak{g}_{\theta}},\mathfrak{z}_{\mathfrak{g}}(\widetilde{\mathfrak{g}_{\theta}}))$	$\Psi\setminus\theta$	Maximal $S$ -Subalgebra?
$F_{4}$	$(A_{1}^{8},G_{2}^{1})$	$\{\alpha_{4}\}$	Yes
$E_{6}$	$(G_{2}^{1},A_{2}^{2^{\prime\prime}})$	$\{\alpha_{2},\alpha_{4}\}$	Yes
$E_{7}$	$(G_{2}^{1},C_{3}^{1^{\prime\prime}})$	$\{\alpha_{1},\alpha_{3}\}$	Yes
	$(A_{1}^{7},G_{2}^{2})$	$\{\alpha_{2}\}$	Yes
	$(A_{1}^{24},A_{1}^{15})$	$\{\alpha_{4}\}$	Yes
	$(A_{1}^{3^{\prime\prime}},F_{4}^{1})$	$\{\alpha_{7}\}$	Yes
	$(A_{1}^{8},G_{2}^{1}\oplus A_{1}^{3^{\prime\prime}})$	$\{\alpha_{6}\}$	No $(\subset F_{4}^{1}\oplus A_{1}^{3^{\prime\prime}})$
$E_{8}$	$(A_{1}^{16},A_{2}^{6^{\prime}})$	$\{\alpha_{2}\}$	Yes
	$(G_{2}^{1},F_{4}^{1})$	$\{\alpha_{7},\alpha_{8}\}$	Yes
	$(A_{1}^{8},G_{2}^{1}\oplus G_{2}^{1})$	$\{\alpha_{1}\}$	No $(\subset G_{2}^{1}\oplus F_{4}^{1})$

Table 8.

S

-irreducible dual pairs in the exceptional Lie algebras.

9.2. Non- $S$ -irreducible dual pairs in the exceptional Lie algebras

The $S$ -irreducible dual pairs in the exceptional Lie algebras are shown in Table 8. To find the non- $S$ -irreducible dual pairs in the exceptional Lie algebras, we will carry out the process described in Section 7.

9.2.1. Dual pairs in $G_{2}$

Recall that $G_{2}$ has no $S$ -irreducible dual pairs (see Table 8). Additionally, recall that by Table 2 and Proposition 15, $(A_{1},\widetilde{A_{1}})$ is a dual pair in $G_{2}$ . By Theorem 28, to find any remaining non- $S$ -irreducible dual pairs, it suffices to consider the proper $S$ -subalgebras of $A_{1}\oplus\widetilde{A_{1}}$ , of which there are none. Therefore, $(A_{1},\widetilde{A_{1}})$ is the only dual pair in $G_{2}$ .

Max-Rank Reg. Subalgebra	Dual Pair	How to Verify
$A_{1}\oplus\widetilde{A_{1}}$	$(A_{1},\widetilde{A_{1}})$	Proposition 15

Table 9. A complete list of dual pairs in

G_{2}

9.2.2. Dual pairs in $F_{4}$

By Table 8, $(A_{1}^{8},G_{2}^{1})$ is an $S$ -irreducible dual pair in $F_{4}$ . Additionally, as indicated in Table 3, we have that $(A_{1},C_{3})$ and $(A_{2},\widetilde{A_{2}})$ are dual pairs in $F_{4}$ . By Theorem 28, to find the remaining non- $S$ -irreducible dual pairs of $F_{4}$ , it suffices to consider the $S$ -subalgebras of the maximal-rank regular subalgebras of $F_{4}$ and to eliminate or confirm the resulting candidate dual pairs. The results of this analysis are summarized in Table 10.

Max-Rank Reg. Subalgebra	Dual Pair	How to Verify
$B_{4}$	$(A_{1}^{6},A_{1}^{6})$	[4, p. 401]
$A_{2}\oplus\widetilde{A_{2}}$	$(A_{2},\widetilde{A_{2}})$	Proposition 15
$C_{3}\oplus A_{1}$	$(C_{3},A_{1})$	Proposition 15
	$(A_{1}^{3},A_{1}^{8}\oplus A_{1})$	Table 6, [4, p. 401]
$A_{3}\oplus\widetilde{A_{1}}$	$(A_{3},\widetilde{A_{1}})$	Example 31
$B_{2}\oplus 2A_{1}$	$(B_{2},2A_{1})$	Table 6, Lemma 33/[5, Table 25]

Table 10. A complete list of the non-

S

-irreducible dual pairs in

F_{4}

9.2.3. Dual pairs in $E_{6}$

Recall from Table 8 that $(G_{2}^{1},A_{2}^{2^{\prime\prime}})$ is an $S$ -irreducible dual pair in $E_{6}$ . Additionally, as indicated in Table 3, we have that $(A_{1},A_{5})$ and $(A_{2},2A_{2})$ are dual pairs in $E_{6}$ . By Theorem 28, to find the remaining non- $S$ -irreducible dual pairs of $E_{6}$ , it suffices to consider the $S$ -subalgebras of the maximal-rank regular subalgebras of $E_{6}$ and to eliminate or confirm the resulting candidate dual pairs. Carrying out this process, we get a complete list of non- $S$ -irreducible dual pairs in $E_{6}$ as shown in Table 11.

Max-Rank Reg. Subalgebra	Dual Pair	How to Verify
$A_{5}\oplus A_{1}$	$(A_{5},A_{1})$	Proposition 15
	$(A_{1}^{3},A_{2}^{2^{\prime\prime}}\oplus A_{1})$	Example 36
$3A_{2}$	$(A_{2},2A_{2})$	Proposition 15

Table 11. A complete list of the non-

S

-irreducible dual pairs in

E_{6}

9.2.4. Dual pairs in $E_{7}$

By Table 8, $(G_{2}^{1},C_{3}^{1^{\prime\prime}})$ , $(A_{1}^{7},G_{2}^{2})$ , $(A_{1}^{24},A_{1}^{15})$ , $(A_{1}^{3^{\prime\prime}},F_{4}^{1})$ , and $(A_{1}^{8},G_{2}^{1}\oplus A_{1}^{3^{\prime\prime}})$ are $S$ -irreducible dual pairs in $E_{7}$ . Additionally, as indicated in Table 3, we have that $(A_{1},D_{6})$ and $(A_{2},A_{5}^{{}^{\prime\prime}})$ are dual pairs in $E_{7}$ . By Theorem 28, to find the remaining non- $S$ -irreducible dual pairs of $E_{7}$ , it suffices to consider the $S$ -subalgebras of the maximal-rank regular subalgebras of $E_{7}$ and to eliminate or confirm the resulting candidate dual pairs. The results of this analysis are summarized in Table 12.

Max-Rank Reg. Subalgebra	Dual Pair	How to Verify
$D_{6}\oplus A_{1}$	$(D_{6},A_{1})$	Proposition 15
	$(B_{2}^{1},B_{3}^{1}\oplus A_{1})$	Table 7, Lemma 33/[5, Table 25]
	$(B_{3}^{1},(B_{2}^{1}\oplus A_{1})^{{}^{\prime\prime}})$	Table 7, Lemma 33/[5, Table 25]
	$(C_{3}^{1^{\prime}},A_{1}^{3^{\prime}}\oplus A_{1})$	Table 7, Remark 37
	$(A_{1}^{3^{\prime}},C_{3}^{1^{\prime}}\oplus A_{1})$	Table 7, Lemma 33/[5, Table 25]
	$(A_{1}^{3^{\prime}}\oplus A_{1}^{3^{\prime\prime}},A_{1}^{8}\oplus A_{1})$	Previous entry, Tables 6 & 7
	$(B_{4}^{1},(A_{1}^{2}\oplus A_{1})^{{}^{\prime\prime}})$	Table 7, Lemma 33/[5, Table 25]
	$(A_{1}^{2},B_{4}^{1}\oplus A_{1})$	Table 7, [4, p. 403]
	$(A_{1}^{6},A_{1}^{6}\oplus A_{1}^{2}\oplus A_{1})$	Table 7, [4, p. 403]
	$(A_{1}^{6}\oplus A_{1}^{2},A_{1}^{6}\oplus A_{1})$	Previous entry, [4, p. 403]
$A_{5}^{\prime\prime}\oplus A_{2}$	$(A_{5}^{\prime\prime},A_{2})$	Proposition 15
	$(A_{2}^{2^{\prime}},A_{1}^{3^{\prime\prime}}\oplus A_{2})$	Example 38
$D_{4}\oplus 3A_{1}$	$(D_{4},3A_{1})$	Table 7, Lemma 33/[5, Table 25]
	$(D_{4}\oplus A_{1},2A_{1})$	Proposition 15 for $D_{6}$
	$((B_{2}^{1}\oplus A_{1})^{{}^{\prime}},A_{1}^{2}\oplus 2A_{1})$	Previous entry, Tables 7 & 19
	$((A_{1}^{2}\oplus A_{1})^{{}^{\prime}},B_{2}^{1}\oplus 2A_{1})$	Table 7, $(D_{4}\oplus A_{1},2A_{1})$
$2A_{3}\oplus A_{1}$	$(A_{3},A_{3}\oplus A_{1})$	Table 7, Lemma 33/[5, Table 25]

Table 12. A complete list of the non-

S

-irreducible dual pairs in

E_{7}

. Here,

(B_{2}^{1}\oplus A_{1})^{{}^{\prime}}

(resp.

(B_{2}^{1}\oplus A_{1})^{{}^{\prime\prime}}

) refers to the subalgebra of

E_{7}

that restricts to

B_{2}^{1^{\prime}}\oplus A_{1}

(resp.

B_{2}^{1^{\prime\prime}}\oplus A_{1}

) in

D_{6}\oplus A_{1}

. Similarly,

(A_{1}^{2}\oplus A_{1})^{{}^{\prime}}

(resp.

(A_{1}^{2}\oplus A_{1})^{{}^{\prime\prime}}

) restricts to

A_{1}^{2^{\prime}}\oplus A_{1}

(resp.

A_{1}^{2^{\prime\prime}}\oplus A_{1}

) in

D_{6}\oplus A_{1}

9.2.5. Dual pairs in $E_{8}$

Finally, recall from Table 8 that $(A_{1}^{16},A_{2}^{6^{\prime}})$ , $(G_{2}^{1},F_{4}^{1})$ , and $(A_{1}^{8},G_{2}^{1}\oplus G_{2}^{1})$ are $S$ -irreducible dual pairs in $E_{8}$ . Additionally, as indicated in Table 3, we have that $(A_{4},A_{4})$ , $(A_{2},E_{6})$ , and $(A_{1},E_{7})$ are dual pairs in $E_{8}$ . By considering the list of admissible subalgebras of $E_{8}$ , we also get that $(A_{1}^{24},A_{1}^{15}\oplus A_{1})$ and $(G_{2}^{2},A_{1}^{7}\oplus A_{1})$ are non- $S$ -irreducible dual pairs [15, Section 6.10]. By Theorem 28, to find the remaining non- $S$ -irreducible dual pairs of $E_{8}$ , it suffices to consider the $S$ -subalgebras of the maximal-rank regular subalgebras of $E_{8}$ and to eliminate or confirm the resulting candidate dual pairs. Carrying out this process, we get the following complete list of non- $S$ -irreducible dual pairs in $E_{8}$ .

Table 13. A complete list of the non-

S

-irreducible dual pairs in

E_{8}

. Here,

(A_{1}^{3}\oplus A_{1})^{{}^{\prime}}

and

(C_{3}^{1}\oplus A_{1})^{{}^{\prime}}

refer to the subalgebras of

E_{8}

that restrict to

A_{1}^{3^{\prime}}

and

C_{3}^{1^{\prime}}

, respectively, in

E_{7}\oplus A_{1}

. Similarly,

(B_{2}^{1}\oplus A_{1}^{2})^{{}^{\prime}}

restricts to

B_{2}^{1^{\prime}}

B_{6}

and to

A_{1}^{2^{\prime}}

B_{5}

Max-Rank Reg. Subalgebra	Dual Pair	How to Verify
$D_{8}$	$(A_{1}^{2},B_{6}^{1})$	[4, p. 405], Lemma 33/[5, Table 25]
	$(A_{1}^{6},A_{1}^{6}\oplus B_{3}^{1})$	[4, p. 405], $(B_{3}^{1},B_{4}^{1})$ , Table 5
	$(A_{1}^{10^{\prime\prime}},B_{2}^{3^{\prime\prime}})$	[4, p. 405], Remark 37
	$(B_{2}^{1},B_{5}^{1})$	Lemma 33/[5, Table 25]
	$(B_{3}^{1},B_{4}^{1})$	Example 35
	$(B_{2}^{2^{\prime\prime}},B_{2}^{2^{\prime\prime}})$	Remark 37
	$(C_{4}^{1^{\prime\prime}},A_{1}^{4^{\prime}})$	[4, p. 405], Remark 37
$E_{7}\oplus A_{1}$	$(E_{7},A_{1})$	Proposition 15
	$(C_{3}^{1},G_{2}^{1}\oplus A_{1})$	Table 8, Lemma 33/[5, Table 25]
	$(A_{1}^{3},F_{4}^{1}\oplus A_{1})$	Table 8, [4, p. 405]
	$(G_{2}^{2},A_{1}^{7}\oplus A_{1})$	[15, Section 6.10]
	$(A_{1}^{7},G_{2}^{2}\oplus A_{1})$	Table 8, [4, p. 405]
	$(A_{1}^{24},A_{1}^{15}\oplus A_{1})$	[15, Section 6.10]
	$(A_{1}^{15},A_{1}^{24}\oplus A_{1})$	Table 8, [4, p. 405]
	$(G_{2}^{1}\oplus A_{1}^{3},A_{1}^{8}\oplus A_{1})$	$(A_{1}^{3},F_{4}^{1}\oplus A_{1})$ , Table 8, [4, p. 403]
$E_{6}\oplus A_{2}$	$(E_{6},A_{2})$	Proposition 15
	$(A_{2}^{2},G_{2}^{1}\oplus A_{2})$	Table 8, Lemma 33/[5, Table 25]
$2A_{4}$	$(A_{4},A_{4})$	Proposition 15
$D_{5}\oplus A_{3}$	$(D_{5},A_{3})$	Lemma 33/[5, Table 25]
$A_{5}\oplus A_{2}\oplus A_{1}$	$(A_{5},A_{2}\oplus A_{1})$	Table 12, Lemma 33/[5, Table 25]
	$(A_{1}^{3}\oplus A_{2},A_{2}^{2}\oplus A_{1})$	Tables 11 & 12
$D_{6}\oplus 2A_{1}$	$(D_{6},2A_{1})$	Table 12, Lemma 33/[5, Table 25]
	$(B_{4}^{1}\oplus A_{1},A_{1}^{2}\oplus A_{1})$	Table 12
	$(B_{3}^{1}\oplus A_{1},B_{2}^{1}\oplus A_{1})$	Table 12
	$(A_{1}^{6}\oplus A_{1}^{2},A_{1}^{6}\oplus 2A_{1})$	$(A_{1}^{2},B_{6}^{1})$ , Tables 5 & 7
	$(A_{1}^{6}\oplus A_{1}^{2}\oplus A_{1},A_{1}^{6}\oplus A_{1})$	Table 12
	$(A_{1}^{8}\oplus 2A_{1},2A_{1}^{3})$	$(A_{1}^{3},F_{4}^{1}\oplus A_{1})$ , Tables 10 & 7
	$((A_{1}^{3}\oplus A_{1})^{{}^{\prime}},(C_{3}^{1}\oplus A_{1})^{{}^{\prime}})$	[4, p. 403], [5, Table 25]
$4A_{2}$	$(2A_{2},2A_{2})$	Table 11
$2D_{4}$	$(D_{4},D_{4})$	Lemma 33/[5, Table 25]
	$((B_{2}^{1}\oplus A_{1}^{2})^{{}^{\prime}},(B_{2}^{1}\oplus A_{1}^{2})^{{}^{\prime}})$	$(A_{1}^{2},B_{6}^{1})$ , $(B_{2}^{1},B_{5}^{1})$ , Table 19
$D_{4}\oplus 4A_{1}$	$(D_{4}\oplus A_{1},3A_{1})$	Tables 12 & 7

Acknowledgements

The author would like to thank David Vogan for suggesting this topic of study and for his helpful comments on the manuscript.

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[17] J. Tits. Algébres alternatives, algébres de Jordan et algébres de Lie exceptionnelles. I. Construction in Nederl. Akad. Wetensch. Proc. Ser. A. Indag. Math., 28:223–237, 1966.

Appendix A: Subalgebras of type $A_{1}$

In this appendix, we describe all of the conjugacy classes of subalgebras of type $A_{1}$ in the complex simple Lie algebras of rank up to 6.⁵⁵5Much of this information is also available in [13], which was used as a reference when compiling these tables. However, while [13] is concerned with $O(n,\mathbb{C})$ conjugacy classes, we are concerned with $SO(n,\mathbb{C})$ conjugacy classes. For example, the subalgebras $A_{1}^{2^{\prime}}$ and $A_{1}^{2^{\prime\prime}}$ of $D_{4}$ appear in two non-conjugate dual pairs of $SO(8,\mathbb{C})$ (see Table 7) but they are not listed as distinct conjugacy classes in [13].

Subalgebras of type $A_{1}$ in $A_{2}$

Index	Defining Vector	Diagram
$1$	$[1,0]$	$[1,1]$
4	$[2,0]$	$[2,2]$

Subalgebras of type $A_{1}$ in $A_{3}$

Index	Defining Vector	Diagram
$1$	$[1,0,0]$	$[1,0,1]$
2	$[1,1,-1]$	$[0,2,0]$
4	$[2,0,0]$	$[2,0,2]$
10	$[3,1,-1]$	$[2,2,2]$

Subalgebras of type $A_{1}$ in $A_{4}$

Index	Defining Vector	Diagram
$1$	$[1,0,0,0]$	$[1,0,0,1]$
2	$[1,1,0,-1]$	$[0,1,1,0]$
4	$[2,0,0,0]$	$[2,0,0,2]$
5	$[2,1,0,-1]$	$[1,1,1,1]$
10	$[3,1,0,-1]$	$[2,1,1,2]$
20	$[4,2,0,-2]$	$[2,2,2,2]$

Subalgebras of type $A_{1}$ in $A_{5}$

Index	Defining Vector	Diagram
$1$	$[1,0,0,0,0]$	$[1,0,0,0,1]$
2	$[1,1,0,0,-1]$	$[0,1,0,1,0]$
3	$[1,1,1,-1,-1]$	$[0,0,2,0,0]$
4	$[2,0,0,0,0]$	$[2,0,0,0,2]$
5	$[2,1,0,0,-1]$	$[1,1,0,1,1]$
8	$[2,2,0,0,-2]$	$[0,2,0,2,0]$
10	$[3,1,0,0,-1]$	$[2,1,0,1,2]$
11	$[3,1,1,-1,-1]$	$[2,0,2,0,2]$
20	$[4,2,0,0,-2]$	$[2,2,0,2,2]$
35	$[5,3,1,-1,-3]$	$[2,2,2,2,2]$

Subalgebras of type $A_{1}$ in $A_{6}$

Index	Defining Vector	Diagram
$1$	$[1,0,0,0,0,0]$	$[1,0,0,0,0,1]$
2	$[1,1,0,0,0,-1]$	$[0,1,0,0,1,0]$
3	$[1,1,1,0,-1,-1]$	$[0,0,1,1,0,0]$
4	$[2,0,0,0,0,0]$	$[2,0,0,0,0,2]$
5	$[2,1,0,0,0,-1]$	$[1,1,0,0,1,1]$
6	$[2,1,1,0,-1,-1]$	$[1,0,1,1,0,1]$
8	$[2,2,0,0,0,-2]$	$[0,2,0,0,2,0]$
10	$[3,1,0,0,0,-1]$	$[2,1,0,0,1,2]$
11	$[3,1,1,0,-1,-1]$	$[2,0,1,1,0,2]$
14	$[3,2,1,0,-1,-2]$	$[1,1,1,1,1,1]$
20	$[4,2,0,0,0,-2]$	$[2,2,0,0,2,2]$
21	$[4,2,1,0,-1,-2]$	$[2,1,1,1,1,2]$
35	$[5,3,1,0,-1,-3]$	$[2,2,1,1,2,2]$
56	$[6,4,2,0,-2,-4]$	$[2,2,2,2,2,2]$

Subalgebras of type $A_{1}$ in $B_{2}$

Index	Defining Vector	Diagram
$1$	$[1,1]$	$[0,1]$
2	$[2,0]$	$[2,0]$
10	$[4,2]$	$[2,2]$

Subalgebras of type $A_{1}$ in $B_{3}$

Index	Defining Vector	Diagram
$1$	$[1,1,0]$	$[0,1,0]$
2	$[2,0,0]$	$[2,0,0]$
3	$[2,1,1]$	$[1,0,1]$
4	$[2,2,0]$	$[0,2,0]$
10	$[4,2,0]$	$[2,2,0]$
28	$[6,4,2]$	$[2,2,2]$

Subalgebras of type $A_{1}$ in $B_{4}$

Index	Defining Vector	Diagram
$1$	$[1,1,0,0]$	$[0,1,0,0]$
$2^{\prime}$	$[1,1,1,1]$	$[0,0,0,1]$
$2^{\prime\prime}$	$[2,0,0,0]$	$[2,0,0,0]$
3	$[2,1,1,0]$	$[1,0,1,0]$
4	$[2,2,0,0]$	$[0,2,0,0]$
6	$[2,2,2,0]$	$[0,0,2,0]$
$10^{\prime}$	$[3,3,1,1]$	$[0,2,0,1]$
$10^{\prime\prime}$	$[4,2,0,0]$	$[2,2,0,0]$
11	$[4,2,1,1]$	$[2,1,0,1]$
12	$[4,2,2,0]$	$[2,0,2,0]$
28	$[6,4,2,0]$	$[2,2,2,0]$
60	$[8,6,4,2]$	$[2,2,2,2]$

Subalgebras of type $A_{1}$ in $B_{5}$

Index	Defining Vector	Diagram
$1$	$[1,1,0,0,0]$	$[0,1,0,0,0]$
$2^{\prime}$	$[1,1,1,1,0]$	$[0,0,0,1,0]$
$2^{\prime\prime}$	$[2,0,0,0,0]$	$[2,0,0,0,0]$
3	$[2,1,1,0,0]$	$[1,0,1,0,0]$
$4^{\prime}$	$[2,1,1,1,1]$	$[1,0,0,0,1]$
$4^{\prime\prime}$	$[2,2,0,0,0]$	$[0,2,0,0,0]$
5	$[2,2,1,1,0]$	$[0,1,0,1,0]$
6	$[2,2,2,0,0]$	$[0,0,2,0,0]$
$10^{\prime}$	$[3,3,1,1,0]$	$[0,2,0,1,0]$
$10^{\prime\prime}$	$[4,2,0,0,0]$	$[2,2,0,0,0]$
11	$[4,2,1,1,0]$	$[2,1,0,1,0]$
$12^{\prime}$	$[3,3,2,1,1]$	$[0,1,1,0,1]$
$12^{\prime\prime}$	$[4,2,2,0,0]$	$[2,0,2,0,0]$
14	$[4,2,2,2,0]$	$[2,0,0,2,0]$
20	$[4,4,2,2,0]$	$[0,2,0,2,0]$
28	$[6,4,2,0,0]$	$[2,2,2,0,0]$
29	$[6,4,2,1,1]$	$[2,2,1,0,1]$
30	$[6,4,2,2,0]$	$[2,2,0,2,0]$
60	$[8,6,4,2,0]$	$[2,2,2,2,0]$
110	$[10,8,6,4,2]$	$[2,2,2,2,2]$

Subalgebras of type $A_{1}$ in $B_{6}$

Index	Defining Vector	Diagram	Index	Defining Vector	Diagram
1	$[1,1,0,0,0,0]$	$[0,1,0,0,0,0]$	$12^{\prime\prime\prime}$	$[4,2,2,0,0,0]$	$[2,0,2,0,0,0]$
$2^{\prime}$	$[1,1,1,1,0,0]$	$[0,0,0,1,0,0]$	13	$[4,2,2,1,1,0]$	$[2,0,1,0,1,0]$
$2^{\prime\prime}$	$[2,0,0,0,0,0]$	$[2,0,0,0,0,0]$	14	$[4,2,2,2,0,0]$	$[2,0,0,2,0,0]$
$3^{\prime}$	$[1,1,1,1,1,1]$	$[0,0,0,0,0,1]$	$20^{\prime}$	$[4,3,3,2,1,1]$	$[1,0,1,1,0,1]$
$3^{\prime\prime}$	$[2,1,1,0,0,0]$	$[1,0,1,0,0,0]$	$20^{\prime\prime}$	$[4,4,2,2,0,0]$	$[0,2,0,2,0,0]$
$4^{\prime}$	$[2,1,1,1,1,0]$	$[1,0,0,0,1,0]$	22	$[4,4,2,2,2,0]$	$[0,2,0,0,2,0]$
$4^{\prime\prime}$	$[2,2,0,0,0,0]$	$[0,2,0,0,0,0]$	28	$[6,4,2,0,0,0]$	$[2,2,2,0,0,0]$
5	$[2,2,1,1,0,0]$	$[0,1,0,1,0,0]$	29	$[6,4,2,1,1,0]$	$[2,2,1,0,1,0]$
6	$[2,2,2,0,0,0]$	$[0,0,2,0,0,0]$	30	$[6,4,2,2,0,0]$	$[2,2,0,2,0,0]$
7	$[2,2,2,1,1,0]$	$[0,0,1,0,1,0]$	32	$[6,4,2,2,2,0]$	$[2,2,0,0,2,0]$
8	$[2,2,2,2,0,0]$	$[0,0,0,2,0,0]$	35	$[5,5,3,3,1,1]$	$[0,2,0,2,0,1]$
$10^{\prime}$	$[3,3,1,1,0,0]$	$[0,2,0,1,0,0]$	38	$[6,4,4,2,2,0]$	$[2,0,2,0,2,0]$
$10^{\prime\prime}$	$[4,2,0,0,0,0]$	$[2,2,0,0,0,0]$	60	$[8,6,4,2,0,0]$	$[2,2,2,2,0,0]$
$11^{\prime}$	$[3,3,1,1,1,1]$	$[0,2,0,0,0,1]$	61	$[8,6,4,2,1,1]$	$[2,2,2,1,0,1]$
$11^{\prime\prime}$	$[4,2,1,1,0,0]$	$[2,1,0,1,0,0]$	62	$[8,6,4,2,2,0]$	$[2,2,2,0,2,0]$
$12^{\prime}$	$[3,3,2,1,1,0]$	$[0,1,1,0,1,0]$	110	$[10,8,6,4,2,0]$	$[2,2,2,2,2,0]$
$12^{\prime\prime}$	$[4,2,1,1,1,1]$	$[2,1,0,0,0,1]$	182	$[12,10,8,6,4,2]$	$[2,2,2,2,2,2]$

Subalgebras of type $A_{1}$ in $C_{2}$

Index	Defining Vector	Diagram
1	$[1,0]$	$[1,0]$
2	$[1,1]$	$[0,2]$
10	$[3,1]$	$[2,2]$

Subalgebras of type $A_{1}$ in $C_{3}$

Index	Defining Vector	Diagram
1	$[1,0,0]$	$[1,0,0]$
2	$[1,1,0]$	$[0,1,0]$
3	$[1,1,1]$	$[0,0,2]$
8	$[2,2,0]$	$[0,2,0]$
10	$[3,1,0]$	$[2,1,0]$
11	$[3,1,1]$	$[2,0,2]$
35	$[5,3,1]$	$[2,2,2]$

Subalgebras of type $A_{1}$ in $C_{4}$

Index	Defining Vector	Diagram
1	$[1,0,0,0]$	$[1,0,0,0]$
2	$[1,1,0,0]$	$[0,1,0,0]$
3	$[1,1,1,0]$	$[0,0,1,0]$
4	$[1,1,1,1]$	$[0,0,0,2]$
8	$[2,2,0,0]$	$[0,2,0,0]$
9	$[2,2,1,0]$	$[0,1,1,0]$
10	$[3,1,0,0]$	$[2,1,0,0]$
11	$[3,1,1,0]$	$[2,0,1,0]$
12	$[3,1,1,1]$	$[2,0,0,2]$
20	$[3,3,1,1]$	$[0,2,0,2]$
35	$[5,3,1,0]$	$[2,2,1,0]$
36	$[5,3,1,1]$	$[2,2,0,2]$
84	$[7,5,3,1]$	$[2,2,2,2]$

Subalgebras of type $A_{1}$ in $C_{5}$

Index	Defining Vector	Diagram
1	$[1,0,0,0,0]$	$[1,0,0,0,0]$
2	$[1,1,0,0,0]$	$[0,1,0,0,0]$
3	$[1,1,1,0,0]$	$[0,0,1,0,0]$
4	$[1,1,1,1,0]$	$[0,0,0,1,0]$
5	$[1,1,1,1,1]$	$[0,0,0,0,2]$
8	$[2,2,0,0,0]$	$[0,2,0,0,0]$
9	$[2,2,1,0,0]$	$[0,1,1,0,0]$
$10^{\prime}$	$[2,2,1,1,0]$	$[0,1,0,1,0]$
$10^{\prime\prime}$	$[3,1,0,0,0]$	$[2,1,0,0,0]$
11	$[3,1,1,0,0]$	$[2,0,1,0,0]$
12	$[3,1,1,1,0]$	$[2,0,0,1,0]$
13	$[3,1,1,1,1]$	$[2,0,0,0,2]$
18	$[3,2,2,1,0]$	$[1,0,1,1,0]$
20	$[3,3,1,1,0]$	$[0,2,0,1,0]$
21	$[3,3,1,1,1]$	$[0,2,0,0,2]$
35	$[5,3,1,0,0]$	$[2,2,1,0,0]$
36	$[5,3,1,1,0]$	$[2,2,0,1,0]$
37	$[5,3,1,1,1]$	$[2,2,0,0,2]$
40	$[4,4,2,2,0]$	$[0,2,0,2,0]$
45	$[5,3,3,1,1]$	$[2,0,2,0,2]$
84	$[7,5,3,1,0]$	$[2,2,2,1,0]$
85	$[7,5,3,1,1]$	$[2,2,2,0,2]$
165	$[9,7,5,3,1]$	$[2,2,2,2,2]$

Subalgebras of type $A_{1}$ in $C_{6}$

Index	Defining Vector	Diagram	Index	Defining Vector	Diagram
1	$[1,0,0,0,0,0]$	$[1,0,0,0,0,0]$	22	$[3,3,1,1,1,1]$	$[0,2,0,0,0,2]$
2	$[1,1,0,0,0,0]$	$[0,1,0,0,0,0]$	30	$[3,3,3,1,1,1]$	$[0,0,2,0,0,2]$
3	$[1,1,1,0,0,0]$	$[0,0,1,0,0,0]$	35	$[5,3,1,0,0,0]$	$[2,2,1,0,0,0]$
4	$[1,1,1,1,0,0]$	$[0,0,0,1,0,0]$	36	$[5,3,1,1,0,0]$	$[2,2,0,1,0,0]$
5	$[1,1,1,1,1,0]$	$[0,0,0,0,1,0]$	37	$[5,3,1,1,1,0]$	$[2,2,0,0,1,0]$
6	$[1,1,1,1,1,1]$	$[0,0,0,0,0,2]$	38	$[5,3,1,1,1,1]$	$[2,2,0,0,0,2]$
8	$[2,2,0,0,0,0]$	$[0,2,0,0,0,0]$	40	$[4,4,2,2,0,0]$	$[0,2,0,2,0,0]$
9	$[2,2,1,0,0,0]$	$[0,1,1,0,0,0]$	41	$[4,4,2,2,1,0]$	$[0,2,0,1,1,0]$
$10^{\prime}$	$[2,2,1,1,0,0]$	$[0,1,0,1,0,0]$	43	$[5,3,2,2,1,0]$	$[2,1,0,1,1,0]$
$10^{\prime\prime}$	$[3,1,0,0,0,0]$	$[2,1,0,0,0,0]$	45	$[5,3,3,1,1,0]$	$[2,0,2,0,1,0]$
$11^{\prime}$	$[2,2,1,1,1,0]$	$[0,1,0,0,1,0]$	46	$[5,3,3,1,1,1]$	$[2,0,2,0,0,2]$
$11^{\prime\prime}$	$[3,1,1,0,0,0]$	$[2,0,1,0,0,0]$	70	$[5,5,3,3,1,1]$	$[0,2,0,2,0,2]$
12	$[3,1,1,1,0,0]$	$[2,0,0,1,0,0]$	84	$[7,5,3,1,0,0]$	$[2,2,2,1,0,0]$
13	$[3,1,1,1,1,0]$	$[2,0,0,0,1,0]$	85	$[7,5,3,1,1,0]$	$[2,2,2,0,1,0]$
14	$[3,1,1,1,1,1]$	$[2,0,0,0,0,2]$	86	$[7,5,3,1,1,1]$	$[2,2,2,0,0,2]$
16	$[2,2,2,2,0,0]$	$[0,0,0,2,0,0]$	94	$[7,5,3,3,1,1]$	$[2,2,0,2,0,2]$
18	$[3,2,2,1,0,0]$	$[1,0,1,1,0,0]$	165	$[9,7,5,3,1,0]$	$[2,2,2,2,1,0]$
19	$[3,2,2,1,1,0]$	$[1,0,1,0,1,0]$	166	$[9,7,5,3,1,1]$	$[2,2,2,2,0,2]$
20	$[3,3,1,1,0,0]$	$[0,2,0,1,0,0]$	286	$[11,9,7,5,3,1]$	$[2,2,2,2,2,2]$
21	$[3,3,1,1,1,0]$	$[0,2,0,0,1,0]$

Subalgebras of type $A_{1}$ in $D_{2}$

Index	Defining Vector	Diagram
$1^{\prime}$	$[1,-1]$	$[2,0]$
$1^{\prime\prime}$	$[1,1]$	$[0,2]$
2	$[2,0]$	$[2,2]$

Subalgebras of type $A_{1}$ in $D_{3}$

Index	Defining Vector	Diagram
1	$[1,1,0]$	$[0,1,1]$
2	$[2,0,0]$	$[2,0,0]$
4	$[2,2,0]$	$[0,2,2]$
10	$[4,2,0]$	$[2,2,2]$

Subalgebras of type $A_{1}$ in $D_{4}$

Index	Defining Vector	Diagram
1	$[1,1,0,0]$	$[0,1,0,0]$
$2^{\prime}$	$[1,1,1,-1]$	$[0,0,2,0]$
$2^{\prime\prime}$	$[1,1,1,1]$	$[0,0,0,2]$
$2^{\prime\prime\prime}$	$[2,0,0,0]$	$[2,0,0,0]$
3	$[2,1,1,0]$	$[1,0,1,1]$
4	$[2,2,0,0]$	$[0,2,0,0]$
$10^{\prime}$	$[3,3,1,-1]$	$[0,2,2,0]$
$10^{\prime\prime}$	$[3,3,1,1]$	$[0,2,0,2]$
$10^{\prime\prime\prime}$	$[4,2,0,0]$	$[2,2,0,0]$
12	$[4,2,2,0]$	$[2,0,2,2]$
28	$[6,4,2,0]$	$[2,2,2,2]$

Subalgebras of type $A_{1}$ in $D_{5}$

Index	Defining Vector	Diagram
1	$[1,1,0,0,0]$	$[0,1,0,0,0]$
$2^{\prime}$	$[1,1,1,1,0]$	$[0,0,0,1,1]$
$2^{\prime\prime}$	$[2,0,0,0,0]$	$[2,0,0,0,0]$
3	$[2,1,1,0,0]$	$[1,0,1,0,0]$
4	$[2,2,0,0,0]$	$[0,2,0,0,0]$
5	$[2,2,1,1,0]$	$[0,1,0,1,1]$
6	$[2,2,2,0,0]$	$[0,0,2,0,0]$
$10^{\prime}$	$[3,3,1,1,0]$	$[0,2,0,1,1]$
$10^{\prime\prime}$	$[4,2,0,0,0]$	$[2,2,0,0,0]$
11	$[4,2,1,1,0]$	$[2,1,0,1,1]$
12	$[4,2,2,0,0]$	$[2,0,2,0,0]$
20	$[4,4,2,2,0]$	$[0,2,0,2,2]$
28	$[6,4,2,0,0]$	$[2,2,2,0,0]$
30	$[6,4,2,2,0]$	$[2,2,0,2,2]$
60	$[8,6,4,2,0]$	$[2,2,2,2,2]$

Subalgebras of type $A_{1}$ in $D_{6}$

Index	Defining Vector	Diagram	Index	Defining Vector	Diagram
1	$[1,1,0,0,0,0]$	$[0,1,0,0,0,0]$	$11^{\prime\prime\prime}$	$[4,2,1,1,0,0]$	$[2,1,0,1,0,0]$
$2^{\prime}$	$[1,1,1,1,0,0]$	$[0,0,0,1,0,0]$	$12^{\prime}$	$[3,3,2,1,1,0]$	$[0,1,1,0,1,1]$
$2^{\prime\prime}$	$[2,0,0,0,0,0]$	$[2,0,0,0,0,0]$	$12^{\prime\prime}$	$[4,2,2,0,0,0]$	$[2,0,2,0,0,0]$
$3^{\prime}$	$[1,1,1,1,1,-1]$	$[0,0,0,0,2,0]$	13	$[4,2,2,1,1,0]$	$[2,0,1,0,1,1]$
$3^{\prime\prime}$	$[1,1,1,1,1,1]$	$[0,0,0,0,0,2]$	14	$[4,2,2,2,0,0]$	$[2,0,0,2,0,0]$
$3^{\prime\prime\prime}$	$[2,1,1,0,0,0]$	$[1,0,1,0,0,0]$	20	$[4,4,2,2,0,0]$	$[0,2,0,2,0,0]$
$4^{\prime}$	$[2,1,1,1,1,0]$	$[1,0,0,0,1,1]$	28	$[6,4,2,0,0,0]$	$[2,2,2,0,0,0]$
$4^{\prime\prime}$	$[2,2,0,0,0,0]$	$[0,2,0,0,0,0]$	29	$[6,4,2,1,1,0]$	$[2,2,1,0,1,1]$
5	$[2,2,1,1,0,0]$	$[0,1,0,1,0,0]$	30	$[6,4,2,2,0,0]$	$[2,2,0,2,0,0]$
6	$[2,2,2,0,0,0]$	$[0,0,2,0,0,0]$	$35^{\prime}$	$[5,5,3,3,1,-1]$	$[0,2,0,2,2,0]$
8	$[2,2,2,2,0,0]$	$[0,0,0,2,0,0]$	$35^{\prime\prime}$	$[5,5,3,3,1,1]$	$[0,2,0,2,0,2]$
$10^{\prime}$	$[3,3,1,1,0,0]$	$[0,2,0,1,0,0]$	38	$[6,4,4,2,2,0]$	$[2,0,2,0,2,2]$
$10^{\prime\prime}$	$[4,2,0,0,0,0]$	$[2,2,0,0,0,0]$	60	$[8,6,4,2,0,0]$	$[2,2,2,2,0,0]$
$11^{\prime}$	$[3,3,1,1,1,-1]$	$[0,2,0,0,2,0]$	62	$[8,6,4,2,2,0]$	$[2,2,2,0,2,2]$
$11^{\prime\prime}$	$[3,3,1,1,1,1]$	$[0,2,0,0,0,2]$	110	$[10,8,6,4,2,0]$	$[2,2,2,2,2,2]$

Appendix B: Subalgebras of ranks 2 and 3

In this appendix, we describe all of the conjugacy classes of subalgebras of ranks 2 and 3 in the complex simple Lie algebras of rank up to 6.⁶⁶6Much of this information is also available in [13], which was used as a reference when compiling these tables. However, while [13] is concerned with $O(n,\mathbb{C})$ conjugacy classes, we are concerned with $SO(n,\mathbb{C})$ conjugacy classes. For example, the subalgebras $B_{2}^{1^{\prime}}$ and $B_{2}^{1^{\prime\prime}}$ of $D_{4}$ appear in two non-conjugate dual pairs of $SO(8,\mathbb{C})$ (see Table 7) but they are not listed as distinct conjugacy classes in [13].

Note that there are some examples where the same conjugacy class can be specified by multiple non-conjugate maps. For example, $A_{2}^{1}$ in $A_{4}$ can be obtained either as the image of $V_{3}\oplus V_{1}^{\oplus 2}$ with Cartan embedding

[x,y,-x-y]\mapsto[x,y,-x-y,0,0]

or as the image of $V_{3}^{*}\oplus V_{1}^{\oplus 2}$ with Cartan embedding

[x,y,-x-y]\mapsto[-x,-y,x+y,0,0].

In these situations, we will include the conjugacy class twice and will include both maps in the table.

For the highest weights and Cartan subalgebras in this appendix, we will use the realizations from [3, Plates I-IX].

Subalgebras of type $A_{2}$

We denote the irreducible representations of $A_{2}$ of dimension at most 13 as follows:

Label	Highest Weight	Dimension	Symplectic or Orthogonal?
$V_{1}$	$[0,0,0]$	1	Orthogonal
$V_{3}$	$\left[\frac{2}{3},-\frac{1}{3},-\frac{1}{3}\right]$	3	Neither
$V_{3}^{*}$	$\left[\frac{1}{3},\frac{1}{3},-\frac{2}{3}\right]$	3	Neither
$V_{\text{Ad}}$	$[1,0,-1]$	8	Orthogonal
$\operatorname{Sym}^{2}V_{3}$	$\left[\frac{4}{3},-\frac{2}{3},-\frac{2}{3}\right]$	6	Neither
$\operatorname{Sym}^{2}V_{3}^{*}$	$\left[\frac{2}{3},\frac{2}{3},-\frac{4}{3}\right]$	6	Neither
$\operatorname{Sym}^{3}V_{3}$	$\left[2,-1,-1\right]$	10	Neither
$\operatorname{Sym}^{3}V_{3}^{*}$	$\left[1,1,-2\right]$	10	Neither

Table 14. Irreducible representations of

A_{2}

of dimension at most 13.

Using this notation, the following table classifies all of the conjugacy classes of subalgebras $\mathfrak{h}$ of type $A_{2}$ in $\mathfrak{g}$ for various $\mathfrak{g}$ .

$\mathfrak{g}$	Dynkin	Standard rep. of	Cartan embedding
	index	$\mathfrak{g}$ restricted to $\mathfrak{h}$
$A_{3}$	$1^{\prime}$	$V_{3}\oplus V_{1}$	$[x,y,-x-y]\mapsto[x,y,-x-y,0]$
	$1^{\prime\prime}$	$V_{3}^{*}\oplus V_{1}$	$[x,y,-x-y]\mapsto[-x,-y,x+y,0]$
$B_{3}$	1	$V_{3}\oplus V_{3}^{*}$	$[x,y,-x-y]\mapsto[x,y,-x-y]$
$C_{3}$	2	$V_{3}\oplus V_{3}^{*}$	$[x,y,-x-y]\mapsto[x,y,-x-y]$
$A_{4}$	1	$V_{3}\oplus V_{1}^{\oplus 2}$	$[x,y,-x-y]\mapsto[x,y,-x-y,0,0]$
	1	$V_{3}^{*}\oplus V_{1}^{\oplus 2}$	$[x,y,-x-y]\mapsto[-x,-y,x+y,0,0]$
$B_{4}$	1	$V_{3}\oplus V_{3}^{*}\oplus V_{1}^{\oplus 3}$	$[x,y,-x-y]\mapsto[x,y,-x-y,0]$
	3	$V_{\text{Ad}}\oplus V_{1}$	$[x,y,-x-y]\mapsto[x-y,x+2y,-2x-y,0]$
$C_{4}$	2	$V_{3}\oplus V_{3}^{*}\oplus V_{1}^{\oplus 2}$	$[x,y,-x-y]\mapsto[x,y,-x-y,0]$
$D_{4}$	1	$V_{3}\oplus V_{3}^{*}\oplus V_{1}^{\oplus 2}$	$[x,y,-x-y]\mapsto[x,y,-x-y,0]$
	3	$V_{\text{Ad}}$	$[x,y,-x-y]\mapsto[x-y,x+2y,-2x-y,0]$
$A_{5}$	1	$V_{3}\oplus V_{1}^{\oplus 3}$	$[x,y,-x-y]\mapsto[x,y,-x-y,0,0,0]$
	1	$V_{3}^{*}\oplus V_{1}^{\oplus 3}$	$[x,y,-x-y]\mapsto[-x,-y,x+y,0,0,0]$
	2	$V_{3}\oplus V_{3}$	$[x,y,-x-y]\mapsto[x,x,y,y,-x-y,-x-y]$
	2	$V_{3}\oplus V_{3}^{*}$	$[x,y,-x-y]\mapsto[x,-x,y,-y,-x-y,x+y]$
	2	$V_{3}^{}\oplus V_{3}^{}$	$[x,y,-x-y]\mapsto[-x,-x,-y,-y,x+y,x+y]$
	5	$\operatorname{Sym}^{2}V_{3}$	$[x,y,-x-y]\mapsto[2x,2y,-2x-2y,-x,-y,x+y]$
	5	$\operatorname{Sym}^{2}V_{3}^{*}$	$[x,y,-x-y]\mapsto[-2x,-2y,2x+2y,x,y,-x-y]$
$B_{5}$	1	$V_{3}\oplus V_{3}^{*}\oplus V_{1}^{\oplus 5}$	$[x,y,-x-y]\mapsto[x,y,-x-y,0,0]$
	3	$V_{\text{Ad}}\oplus V_{1}^{\oplus 3}$	$[x,y,-x-y]\mapsto[x-y,x+2y,-2x-y,0,0]$
$C_{5}$	2	$V_{3}\oplus V_{3}^{*}\oplus V_{1}^{\oplus 4}$	$[x,y,-x-y]\mapsto[x,y,-x-y,0,0]$
$D_{5}$	1	$V_{3}\oplus V_{3}^{*}\oplus V_{1}^{\oplus 4}$	$[x,y,-x-y]\mapsto[x,y,-x-y,0,0]$
	3	$V_{\text{Ad}}\oplus V_{1}^{\oplus 2}$	$[x,y,-x-y]\mapsto[x-y,x+2y,-2x-y,0,0]$
$A_{6}$	1	$V_{3}\oplus V_{1}^{\oplus 4}$	$[x,y,-x-y]\mapsto[x,y,-x-y,0,0,0,0]$
	1	$V_{3}^{*}\oplus V_{1}^{\oplus 4}$	$[x,y,-x-y]\mapsto[-x,-y,x+y,0,0,0,0]$
	2	$V_{3}\oplus V_{3}\oplus V_{1}$	$[x,y,-x-y]\mapsto[x,x,y,y,-x-y,-x-y,0]$
	2	$V_{3}\oplus V_{3}^{*}\oplus V_{1}$	$[x,y,-x-y]\mapsto[x,-x,y,-y,-x-y,x+y,0]$
	2	$V_{3}^{}\oplus V_{3}^{}\oplus V_{1}$	$[x,y,-x-y]\mapsto[-x,-x,-y,-y,x+y,x+y,0]$
	5	$\operatorname{Sym}^{2}V_{3}\oplus V_{1}$	$[x,y,-x-y]\mapsto[2x,2y,-2x-2y,-x,-y,x+y,0]$
	5	$\operatorname{Sym}^{2}V_{3}^{*}\oplus V_{1}$	$[x,y,-x-y]\mapsto[-2x,-2y,2x+2y,x,y,-x-y,0]$
$B_{6}$	1	$V_{3}\oplus V_{3}^{*}\oplus V_{1}^{\oplus 7}$	$[x,y,-x-y]\mapsto[x,y,-x-y,0,0,0]$
	2	$V_{3}\oplus V_{3}\oplus V_{3}^{}\oplus V_{3}^{}\oplus V_{1}$	$[x,y,-x-y]\mapsto[x,x,y,y,-x-y,-x-y]$
	3	$V_{\text{Ad}}\oplus V_{1}^{\oplus 5}$	$[x,y,-x-y]\mapsto[x-y,x+2y,-2x-y,0,0,0]$
	5	$\operatorname{Sym}^{2}V_{3}\oplus\operatorname{Sym}^{2}V_{3}^{*}\oplus V_{1}$	$[x,y,-x-y]\mapsto[2x,2y,-2x-2y,-x,-y,x+y]$
$C_{6}$	2	$V_{3}\oplus V_{3}^{*}\oplus V_{1}^{\oplus 6}$	$[x,y,-x-y]\mapsto[x,y,-x-y,0,0,0]$
	4	$V_{3}\oplus V_{3}\oplus V_{3}^{}\oplus V_{3}^{}$	$[x,y,-x-y]\mapsto[x,x,y,y,-x-y,-x-y]$
	10	$\operatorname{Sym}^{2}V_{3}\oplus\operatorname{Sym}^{2}V_{3}^{*}$	$[x,y,-x-y]\mapsto[2x,2y,-2x-2y,-x,-y,x+y]$

Table 16. Conjugacy classes of subalgebras of type

A_{2}

in various

\mathfrak{g}

Subalgebras of type $B_{2}\simeq C_{2}$

We denote the irreducible representations of $B_{2}\simeq C_{2}$ of dimension at most 13 as follows:

Label	Highest Weight	Highest Weight	Dimension	Symplectic or
	( $B_{2}$ coords.)	( $C_{2}$ coords.)		Orthogonal?
$V_{1}$	$[0,0]$	$[0,0]$	1	Orthogonal
$V_{4}$	$\left[\frac{1}{2},\frac{1}{2}\right]$	$[1,0]$	4	Symplectic
$V_{5}$	$[1,0]$	$[1,1]$	5	Orthogonal
$V_{10}$	$[1,1]$	$[2,0]$	10	Orthogonal

Table 17. Irreducible representations of

B_{2}\simeq C_{2}

of dimension at most 13.

Using this notation, the following table classifies all of the conjugacy classes of subalgebras $\mathfrak{h}$ of type $B_{2}\simeq C_{2}$ in $\mathfrak{g}$ for various $\mathfrak{g}$ .

$\mathfrak{g}$	Dynkin	Standard rep. of	Cartan embedding	Cartan embedding
	index	$\mathfrak{g}$ restricted to $\mathfrak{h}$	( $B_{2}$ coords.)	( $C_{2}$ coords.)
$A_{3}$	1	$V_{4}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x-y}{2},-\frac{x+y}{2},-\frac{x-y}{2}\right]$	$[p,q]\mapsto[p,q,-p,-q]$
$B_{3}$	1	$V_{5}\oplus V_{1}^{\oplus 2}$	$[x,y]\mapsto[x,y,0]$	$[p,q]\mapsto[p+q,p-q,0]$
$C_{3}$	1	$V_{4}\oplus V_{1}^{\oplus 2}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x-y}{2},0\right]$	$[p,q]\mapsto[p,q,0]$
$D_{3}$	1	$V_{5}\oplus V_{1}$	$[x,y]\mapsto[x,y,0]$	$[p,q]\mapsto[p+q,p-q,0]$
$A_{4}$	1	$V_{4}\oplus V_{1}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x-y}{2},-\frac{x+y}{2},-\frac{x-y}{2},0\right]$	$[p,q]\mapsto[p,q,-p,-q,0]$
	2	$V_{5}$	$[x,y]\mapsto[x,y,-x,-y,0]$	$[p,q]\mapsto[p+q,p-q,-p-q,q-p,0]$
$B_{4}$	$1^{\prime}$	$V_{4}\oplus V_{4}\oplus V_{1}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x+y}{2},\frac{x-y}{2},\frac{x-y}{2}\right]$	$[p,q]\mapsto[p,p,q,q]$
	$1^{\prime\prime}$	$V_{5}\oplus V_{1}^{\oplus 4}$	$[x,y]\mapsto[x,y,0,0]$	$[p,q]\mapsto\left[p+q,p-q,0,0\right]$
$C_{4}$	1	$V_{4}\oplus V_{1}^{\oplus 4}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x-y}{2},0,0\right]$	$[p,q]\mapsto[p,q,0,0]$
	2	$V_{4}\oplus V_{4}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x+y}{2},\frac{x-y}{2},\frac{x-y}{2}\right]$	$[p,q]\mapsto[p,p,q,q]$
$D_{4}$	$1^{\prime}$	$V_{4}\oplus V_{4}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x+y}{2},\frac{x-y}{2},-\frac{x-y}{2}\right]$	$[p,q]\mapsto[p,p,q,-q]$
	$1^{\prime\prime}$	$V_{4}\oplus V_{4}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x+y}{2},\frac{x-y}{2},\frac{x-y}{2}\right]$	$[p,q]\mapsto[p,p,q,q]$
	$1^{\prime\prime\prime}$	$V_{5}\oplus V_{1}^{\oplus 3}$	$[x,y]\mapsto[x,y,0,0]$	$[p,q]\mapsto[p+q,p-q,0,0]$
$A_{5}$	1	$V_{4}\oplus V_{1}^{\oplus 2}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x-y}{2},-\frac{x+y}{2},-\frac{x-y}{2},0,0\right]$	$[p,q]\mapsto[p,q,-p,-q,0,0]$
	2	$V_{5}\oplus V_{1}$	$[x,y]\mapsto[x,y,-x,-y,0,0]$	$[p,q]\mapsto[p+q,p-q,-p-q,q-p,0,0]$
$B_{5}$	$1^{\prime}$	$V_{4}\oplus V_{4}\oplus V_{1}^{\oplus 3}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x+y}{2},\frac{x-y}{2},\frac{x-y}{2},0\right]$	$[p,q]\mapsto[p,p,q,q,0]$
	$1^{\prime\prime}$	$V_{5}\oplus V_{1}^{\oplus 6}$	$[x,y]\mapsto[x,y,0,0,0]$	$[p,q]\mapsto[p+q,p-q,0,0,0]$
	2	$V_{5}\oplus V_{5}\oplus V_{1}$	$[x,y]\mapsto[x,x,y,y,0]$	$[p,q]\mapsto[p+q,p+q,p-q,p-q,0]$
	3	$V_{10}\oplus V_{1}$	$[x,y]\mapsto[x,y,x+y,x-y,0]$	$[p,q]\mapsto[p+q,p-q,2p,2q,0]$
$C_{5}$	1	$V_{4}\oplus V_{1}^{\oplus 6}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x-y}{2},0,0,0\right]$	$[p,q]\mapsto[p,q,0,0,0]$
	2	$V_{4}\oplus V_{4}\oplus V_{1}^{\oplus 2}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x+y}{2},\frac{x-y}{2},\frac{x-y}{2},0\right]$	$[p,q]\mapsto[p,p,q,q,0]$
	4	$V_{5}\oplus V_{5}$	$[x,y]\mapsto[x,x,y,y,0]$	$[p,q]\mapsto[p+q,p+q,p-q,p-q,0]$
$D_{5}$	$1^{\prime}$	$V_{4}\oplus V_{4}\oplus V_{1}^{\oplus 2}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x+y}{2},\frac{x-y}{2},\frac{x-y}{2},0\right]$	$[p,q]\mapsto[p,p,q,q,0]$
	$1^{\prime\prime}$	$V_{5}\oplus V_{1}^{\oplus 5}$	$[x,y]\mapsto[x,y,0,0,0]$	$[p,q]\mapsto[p+q,p-q,0,0,0]$
	2	$V_{5}\oplus V_{5}$	$[x,y]\mapsto[x,x,y,y,0]$	$[p,q]\mapsto[p+q,p+q,p-q,p-q,0]$
	3	$V_{10}$	$[x,y]\mapsto[x,y,x+y,x-y,0]$	$[p,q]\mapsto[p+q,p-q,2p,2q,0]$
$A_{6}$	1	$V_{4}\oplus V_{1}^{\oplus 3}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x-y}{2},-\frac{x+y}{2},-\frac{x-y}{2},0,0,0\right]$	$[p,q]\mapsto[p,q,-p,-q,0,0,0]$
	2	$V_{5}\oplus V_{1}^{\oplus 2}$	$[x,y]\mapsto[x,y,-x,-y,0,0,0]$	$[p,q]\mapsto[p+q,p-q,-p-q,q-p,0,0,0]$
$B_{6}$	$1^{\prime}$	$V_{4}\oplus V_{4}\oplus V_{1}^{\oplus 5}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x+y}{2},\frac{x-y}{2},\frac{x-y}{2},0,0\right]$	$[p,q]\mapsto[p,p,q,q,0,0]$
	$1^{\prime\prime}$	$V_{5}\oplus V_{1}^{\oplus 8}$	$[x,y]\mapsto[x,y,0,0,0,0]$	$[p,q]\mapsto[p+q,p-q,0,0,0,0]$
	$2^{\prime}$	$V_{5}\oplus V_{5}\oplus V_{1}^{\oplus 3}$	$[x,y]\mapsto[x,x,y,y,0,0]$	$[p,q]\mapsto[p+q,p+q,p-q,p-q,0,0]$
	$2^{\prime\prime}$	$V_{5}\oplus V_{4}\oplus V_{4}$	$[x,y]\mapsto\left[x,y,\frac{x+y}{2},\frac{x+y}{2},\frac{x-y}{2},\frac{x-y}{2}\right]$	$[p,q]\mapsto[p+q,p-q,p,p,q,q]$
	3	$V_{10}\oplus V_{1}^{\oplus 3}$	$[x,y]\mapsto[x,y,x+y,x-y,0,0]$	$[p,q]\mapsto[p+q,p-q,2p,2q,0,0]$
$C_{6}$	1	$V_{4}\oplus V_{1}^{\oplus 8}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x-y}{2},0,0,0,0\right]$	$[p,q]\mapsto[p,q,0,0,0,0]$
	2	$V_{4}\oplus V_{4}\oplus V_{1}^{\oplus 4}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x+y}{2},\frac{x-y}{2},\frac{x-y}{2},0,0\right]$	$[p,q]\mapsto[p,p,q,q,0,0]$
	3	$V_{4}\oplus V_{4}\oplus V_{4}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x+y}{2},\frac{x+y}{2},\frac{x-y}{2},\frac{x-y}{2},\frac{x-y}{2}\right]$	$[p,q]\mapsto[p,p,p,q,q,q]$
	4	$V_{5}\oplus V_{5}\oplus V_{1}^{\oplus 2}$	$[x,y]\mapsto[x,x,y,y,0,0]$	$[p,q]\mapsto[p+q,p+q,p-q,p-q,0,0]$
$D_{6}$	$1^{\prime}$	$V_{4}\oplus V_{4}\oplus V_{1}^{\oplus 4}$	$[x,y]\mapsto\left[\frac{x+y}{2},\frac{x+y}{2},\frac{x-y}{2},\frac{x-y}{2},0,0\right]$	$[p,q]\mapsto[p,p,q,q,0,0]$
	$1^{\prime\prime}$	$V_{5}\oplus V_{1}^{\oplus 7}$	$[x,y]\mapsto[x,y,0,0,0,0]$	$[p,q]\mapsto[p+q,p-q,0,0,0,0]$
	2	$V_{5}\oplus V_{5}\oplus V_{1}^{\oplus 2}$	$[x,y]\mapsto[x,x,y,y,0,0]$	$[p,q]\mapsto[p+q,p+q,p-q,p-q,0,0]$
	3	$V_{10}\oplus V_{1}^{\oplus 2}$	$[x,y]\mapsto[x,y,x+y,x-y,0,0]$	$[p,q]\mapsto[p+q,p-q,2p,2q,0,0]$

Table 19. Conjugacy classes of subalgebras of type

B_{2}\simeq C_{2}

in various

\mathfrak{g}

Subalgebras of type $G_{2}$

We denote the irreducible representations of $G_{2}$ of dimension at most 13 as follows:

Label	Highest Weight	Dimension	Symplectic or Orthogonal?
$V_{1}$	$[0,0,0]$	1	Orthogonal
$V_{7}$	$\left[0,-1,1\right]$	7	Orthogonal

Table 20. Irreducible representations of

G_{2}

of dimension at most 13.

Using this notation, the following table classifies all of the conjugacy classes of subalgebras $\mathfrak{h}$ of type $G_{2}$ in $\mathfrak{g}$ for various $\mathfrak{g}$ .

$\mathfrak{g}$	Dynkin	Standard rep. of	Cartan embedding
	index	$\mathfrak{g}$ restricted to $\mathfrak{h}$
$B_{3}$	1	$V_{7}$	$[x,y,-x-y]\mapsto[x-y,2x+y,x+2y]$
$B_{4}$	1	$V_{7}\oplus V_{1}^{\oplus 2}$	$[x,y,-x-y]\mapsto[x-y,2x+y,x+2y,0]$
$D_{4}$	1	$V_{7}\oplus V_{1}$	$[x,y,-x-y]\mapsto[x-y,2x+y,x+2y,0]$
$B_{5}$	1	$V_{7}\oplus V_{1}^{\oplus 4}$	$[x,y,-x-y]\mapsto[x-y,2x+y,x+2y,0,0]$
$D_{5}$	1	$V_{7}\oplus V_{1}^{\oplus 3}$	$[x,y,-x-y]\mapsto[x-y,2x+y,x+2y,0,0]$
$A_{6}$	2	$V_{7}$	$[x,y,-x-y]\mapsto[x-y,2x+y,x+2y,y-x,-2x-y,-x-2y,0]$
$B_{6}$	1	$V_{7}\oplus V_{1}^{\oplus 6}$	$[x,y,-x-y]\mapsto[x-y,2x+y,x+2y,0,0,0]$
$D_{6}$	1	$V_{7}\oplus V_{1}^{\oplus 5}$	$[x,y,-x-y]\mapsto[x-y,2x+y,x+2y,0,0,0]$

Table 22. Conjugacy classes of subalgebras of type

G_{2}

in various

\mathfrak{g}

Subalgebras of type $A_{3}\simeq D_{3}$

We denote the irreducible representations of $A_{3}\simeq D_{3}$ of dimension at most 13 as follows:

Label	Highest Weight	Highest Weight	Dimension	Symplectic or
	( $A_{3}$ coords.)	( $D_{3}$ coords.)		Orthogonal?
$V_{1}$	$[0,0,0,0]$	$[0,0,0]$	1	Orthogonal
$V_{4}$	$\left[\frac{3}{4},-\frac{1}{4},-\frac{1}{4},-\frac{1}{4}\right]$	$\left[\frac{1}{2},\frac{1}{2},\frac{1}{2}\right]$	4	Neither
$V_{4}^{*}$	$\left[\frac{1}{4},\frac{1}{4},\frac{1}{4},-\frac{3}{4}\right]$	$\left[\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right]$	4	Neither
$\bigwedge^{2}V_{4}$	$\left[\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right]$	$[1,0,0]$	6	Orthogonal
$\operatorname{Sym}^{2}V_{4}$	$\left[\frac{3}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right]$	$[1,1,1]$	10	Neither
$\operatorname{Sym}^{2}V_{4}^{*}$	$\left[\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{3}{2}\right]$	$[1,1,-1]$	10	Neither

Table 23. Irreducible representations of

A_{3}\simeq D_{3}

of dimension at most 13. (Note that

\bigwedge^{2}V_{4}

can also be viewed as the standard representation of

D_{3}

Using this notation, the following table classifies all of the conjugacy classes of subalgebras $\mathfrak{h}$ of type $A_{3}\simeq D_{3}$ in $\mathfrak{g}$ for various $\mathfrak{g}$ .

$\mathfrak{g}$	Dynkin	Standard rep. of	Cartan embedding
	index	$\mathfrak{g}$ restricted to $\mathfrak{h}$
$B_{3}$	1	$\bigwedge^{2}V_{4}\oplus V_{1}$	$[x,y,z,-x-y-z]\mapsto[x-z,y-z,x+y]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto[p,q,r]$	( $D_{3}$ coords.)
$A_{4}$	1	$V_{4}\oplus V_{1}$	$[x,y,z,-x-y-z]\mapsto[x,y,z,-x-y-z,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[-p+q-r,p-q-r,-p-q+r,p+q+r,0\right]$	( $D_{3}$ coords.)
	1	$V_{4}^{*}\oplus V_{1}$	$[x,y,z,-x-y-z]\mapsto[-x,-y,-z,x+y+z,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[p-q+r,-p+q+r,p+q-r,-p-q-r,0\right]$	( $D_{3}$ coords.)
$B_{4}$	$1^{\prime}$	$V_{4}\oplus V_{4}^{*}\oplus V_{1}$	$[x,y,z,-x-y-z]\mapsto[x,y,z,-x-y-z]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[-p+q-r,p-q-r,-p-q+r,p+q+r\right]$	( $D_{3}$ coords.)
	$1^{\prime\prime}$	$\bigwedge^{2}V_{4}\oplus V_{1}^{\oplus 3}$	$[x,y,z,-z-y-z]\mapsto[x-z,y-z,x+y,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto[p,q,r,0]$	( $D_{3}$ coords.)
$C_{4}$	2	$V_{4}\oplus V_{4}^{*}$	$[x,y,z,-x-y-z]\mapsto[x,y,z,-x-y-z]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[-p+q-r,p-q-r,-p-q+r,p+q+r\right]$	( $D_{3}$ coords.)
$D_{4}$	$1^{\prime}$	$V_{4}\oplus V_{4}^{*}$	$[x,y,z,-x-y-z]\mapsto[x,y,z,-x-y-z]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[-p+q-r,p-q-r,-p-q+r,p+q+r\right]$	( $D_{3}$ coords.)
	$1^{\prime\prime}$	$\bigwedge^{2}V_{4}\oplus V_{1}^{\oplus 2}$	$[x,y,z,-x-y-z]\mapsto[x-z,y-z,x+y,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto[p,q,r,0]$	( $D_{3}$ coords.)
$A_{5}$	1	$V_{4}\oplus V_{1}^{\oplus 2}$	$[x,y,z,-x-y-z]\mapsto[x,y,z,-x-y-z,0,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[-p+q-r,p-q-r,-p-q+r,p+q+r,0,0\right]$	( $D_{3}$ coords.)
	1	$V_{4}^{*}\oplus V_{1}^{\oplus 2}$	$[x,y,z,-x-y-z]\mapsto[-x,-y,-z,x+y+z,0,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[p-q+r,-p+q+r,p+q-r,-p-q-r,0,0\right]$	( $D_{3}$ coords.)
	2	$\bigwedge^{2}V_{4}$	$[x,y,z,-x-y-z]\mapsto[x-z,y-z,x+y,-x+z,-y+z,-x-y]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto[p,q,r,-p,-q,-r]$	( $D_{3}$ coords.)
$B_{5}$	$1^{\prime}$	$V_{4}\oplus V_{4}^{*}\oplus V_{1}^{\oplus 3}$	$[x,y,z,-x-y-z]\mapsto[x,y,z,-x-y-z,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[-p+q-r,p-q-r,-p-q+r,p+q+r,0\right]$	( $D_{3}$ coords.)
	$1^{\prime\prime}$	$\bigwedge^{2}V_{4}\oplus V_{1}^{\oplus 5}$	$[x,y,z,-x-y-z]\mapsto[x-z,y-z,x+y,0,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto[p,q,r,0,0]$	( $D_{3}$ coords.)
$C_{5}$	2	$V_{4}\oplus V_{4}^{*}\oplus V_{1}^{\oplus 2}$	$[x,y,z,-x-y-z]\mapsto[x,y,z,-x-y-z,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[-p+q-r,p-q-r,-p-q+r,p+q+r,0\right]$	( $D_{3}$ coords.)
$D_{5}$	$1^{\prime}$	$V_{4}\oplus V_{4}^{*}\oplus V_{1}^{\oplus 2}$	$[x,y,z,-x-y-z]\mapsto[x,y,z,-x-y-z,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[-p+q-r,p-q-r,-p-q+r,p+q+r,0\right]$	( $D_{3}$ coords.)
	$1^{\prime\prime}$	$\bigwedge^{2}V_{4}\oplus V_{1}^{\oplus 4}$	$[x,y,z,-x-y-z]\mapsto[x-z,y-z,x+y,0,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto[p,q,r,0,0]$	( $D_{3}$ coords.)
$A_{6}$	1	$V_{4}\oplus V_{1}^{\oplus 3}$	$[x,y,z,-x-y-z]\mapsto[x,y,z,-x-y-z,0,0,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[-p+q-r,p-q-r,-p-q+r,p+q+r,0,0,0\right]$	( $D_{3}$ coords.)
	1	$V_{4}^{*}\oplus V_{1}^{\oplus 3}$	$[x,y,z,-x-y-z]\mapsto[-x,-y,-z,x+y+z,0,0,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[p-q+r,-p+q+r,p+q-r,-p-q-r,0,0,0\right]$	( $D_{3}$ coords.)
	2	$\bigwedge^{2}V_{4}\oplus V_{1}$	$[x,y,z,-x-y-z]\mapsto[x-z,y-z,x+y,-x+z,-y+z,-x-y,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto[p,q,r,-p,-q,-r,0]$	( $D_{3}$ coords.)
$B_{6}$	$1^{\prime}$	$V_{4}\oplus V_{4}^{*}\oplus V_{1}^{\oplus 5}$	$[x,y,z,-x-y-z]\mapsto[x,y,z,-x-y-z,0,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[-p+q-r,p-q-r,-p-q+r,p+q+r,0,0\right]$	( $D_{3}$ coords.)
	$1^{\prime\prime}$	$\bigwedge^{2}V_{4}\oplus V_{1}^{\oplus 7}$	$[x,y,z,-x-y-z]\mapsto[x-z,y-z,x+y,0,0,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto[p,q,r,0,0,0]$	( $D_{3}$ coords.)
	2	$\bigwedge^{2}V_{4}\oplus\bigwedge^{2}V_{4}\oplus V_{1}$	$[x,y,z,-x-y-z]\mapsto[x-z,x-z,y-z,y-z,x+y,x+y]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto[p,p,q,q,r,r]$	( $D_{3}$ coords.)
$C_{6}$	2	$V_{4}\oplus V_{4}^{*}\oplus V_{1}^{\oplus 4}$	$[x,y,z,-x-y-z]\mapsto[x,y,z,-x-y-z,0,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[-p+q-r,p-q-r,-p-q+r,p+q+r,0,0\right]$	( $D_{3}$ coords.)
	4	$\bigwedge^{2}V_{4}\oplus\bigwedge^{2}V_{4}$	$[x,y,z,-x-y-z]\mapsto[x-z,x-z,y-z,y-z,x+y,x+y]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto[p,p,q,q,r,r]$	( $D_{3}$ coords.)
$D_{6}$	$1^{\prime}$	$V_{4}\oplus V_{4}^{*}\oplus V_{1}^{\oplus 4}$	$[x,y,z,-x-y-z]\mapsto[x,y,z,-x-y-z,0,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto\frac{1}{2}\left[-p+q-r,p-q-r,-p-q+r,p+q+r,0,0\right]$	( $D_{3}$ coords.)
	$1^{\prime\prime}$	$\bigwedge^{2}V_{4}\oplus V_{1}^{\oplus 6}$	$[x,y,z,-x-y-z]\mapsto[x-z,y-z,x+y,0,0,0]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto[p,q,r,0,0,0]$	( $D_{3}$ coords.)
	2	$\bigwedge^{2}V_{4}\oplus\bigwedge^{2}V_{4}$	$[x,y,z,-x-y-z]\mapsto[x-z,x-z,y-z,y-z,x+y,x+y]$	( $A_{3}$ coords.)
			$[p,q,r]\mapsto[p,p,q,q,r,r]$	( $D_{3}$ coords.)

Table 25. Conjugacy classes of subalgebras of type

A_{3}\simeq D_{3}

in various

\mathfrak{g}

Subalgebras of type $B_{3}$

We denote the irreducible representations of $B_{3}$ of dimension at most 13 as follows:

Label	Highest Weight	Dimension	Symplectic or Orthogonal?
$V_{1}$	$[0,0,0]$	1	Orthogonal
$V_{7}$	$\left[1,0,0\right]$	7	Orthogonal

Table 26. Irreducible representations of

B_{3}

of dimension at most 13.

Using this notation, the following table classifies all of the conjugacy classes of subalgebras $\mathfrak{h}$ of type $B_{3}$ in $\mathfrak{g}$ for various $\mathfrak{g}$ .

$\mathfrak{g}$	Dynkin	Standard rep. of	Cartan embedding
	index	$\mathfrak{g}$ restricted to $\mathfrak{h}$
$B_{4}$	1	$V_{7}\oplus V_{1}^{\oplus 2}$	$[x,y,z]\mapsto[x,y,z,0]$
$D_{4}$	1	$V_{7}\oplus V_{1}$	$[x,y,z]\mapsto[x,y,z,0]$
$B_{5}$	1	$V_{7}\oplus V_{1}^{\oplus 4}$	$[x,y,z]\mapsto[x,y,z,0,0]$
$D_{5}$	1	$V_{7}\oplus V_{1}^{\oplus 3}$	$[x,y,z]\mapsto[x,y,z,0,0]$
$A_{6}$	2	$V_{7}$	$[x,y,z]\mapsto[x,y,z,-x,-y,-z,0]$
$B_{6}$	1	$V_{7}\oplus V_{1}^{\oplus 6}$	$[x,y,z]\mapsto[x,y,z,0,0,0]$
$D_{6}$	1	$V_{7}\oplus V_{1}^{\oplus 5}$	$[x,y,z]\mapsto[x,y,z,0,0,0]$

Table 28. Conjugacy classes of subalgebras of type

B_{3}

in various

\mathfrak{g}

Subalgebras of type $C_{3}$

We denote the irreducible representations of $C_{3}$ of dimension at most 13 as follows:

Label	Highest Weight	Dimension	Symplectic or Orthogonal?
$V_{1}$	$[0,0,0]$	1	Orthogonal
$V_{6}$	$\left[1,0,0\right]$	6	Symplectic

Table 29. Irreducible representations of

C_{3}

of dimension at most 13.

Using this notation, the following table classifies all of the conjugacy classes of subalgebras $\mathfrak{h}$ of type $C_{3}$ in $\mathfrak{g}$ for various $\mathfrak{g}$ .

$\mathfrak{g}$	Dynkin	Standard rep. of	Cartan embedding
	index	$\mathfrak{g}$ restricted to $\mathfrak{h}$
$C_{4}$	1	$V_{6}\oplus V_{1}^{\oplus 2}$	$[p,q,r]\mapsto[p,q,r,0]$
$A_{5}$	1	$V_{6}$	$[p,q,r]\mapsto[p,q,r,-p,-q,-r]$
$C_{5}$	1	$V_{6}\oplus V_{1}^{\oplus 4}$	$[p,q,r]\mapsto[p,q,r,0,0]$
$A_{6}$	1	$V_{6}\oplus V_{1}$	$[p,q,r]\mapsto[p,q,r,-p,-q,-r,0]$
$B_{6}$	1	$V_{6}\oplus V_{6}\oplus V_{1}$	$[p,q,r]\mapsto[p,p,q,q,r,r]$
$C_{6}$	1	$V_{6}\oplus V_{1}^{\oplus 6}$	$[p,q,r]\mapsto[p,q,r,0,0,0]$
	2	$V_{6}\oplus V_{6}$	$[p,q,r]\mapsto[p,p,q,q,r,r]$
$D_{6}$	$1^{\prime}$	$V_{6}\oplus V_{6}$	$[p,q,r]\mapsto[p,p,q,q,r,-r]$
	$1^{\prime\prime}$	$V_{6}\oplus V_{6}$	$[p,q,r]\mapsto[p,p,q,q,r,r]$

Table 31. Conjugacy classes of subalgebras of type

C_{3}

in various

\mathfrak{g}

An explicit classification of dual pairs in exceptional Lie algebras

Abstract.

1. Introduction

2. Preliminaries

Definition 1 ([9, p. 550]).

2.1. Why we are focusing on semisimple dual pairs in simple Lie algebras

Lemma 2 ([15, Lemma 5.1]).

Definition 3 ([15, Section 1.2]).

Lemma 4 ([15, Lemma 5.2]).

Lemma 5 ([15, Lemma 5.3]).

2.2. Defining a normalized symmetric form on 𝔤\mathfrak{g}

2.3. Dynkin’s index notation

Example 6.

2.4. Dynkin’s defining vector conventions and weighted diagrams

Remark 7.

Example 8.

3. Maximal and maximal-rank regular subalgebras

Definition 9 ([15, Definition 5.6]).

Definition 10.

3.1. Maximal-rank regular subalgebras

Remark 11.

Example 12.

3.2. Maximal regular subalgebras of maximal rank

Lemma 13.

Proof.

Example 14.

3.3. Dual pairs from maximal regular subalgebras of maximal rank

Proposition 15 ([15, Proposition 5.15]).

4. Admissible subalgebras

Definition 16 ([15, Definition 3.1]).

Remark 17.

Theorem 18 ([15, Theorem 2.3]).

Remark 19.

Theorem 20 ([15, Theorem 3.3]).

Example 21.

5. SS-irreducible and admissible dual pairs

Definition 22 ([15, Definition 5.9]).

Definition 23 ([15, Definition 5.11]).

Remark 24.

Proposition 25 ([15, Proposition 5.19]).

Example 26.

Theorem 27 ([15, Theorem 8.1]).

6. Some of the dual pairs in the classical simple Lie algebras

6.1. Type AnA_{n}

6.2. Type BnB_{n}

6.3. Type CnC_{n}

6.4. Type DnD_{n}

7. Non-SS-irreducible dual pairs

Theorem 28 ([15, Theorem 5.12]).

Lemma 29.

Proof of (a).

Proof of (b).

8. Eliminating and confirming candidate dual pairs

8.1. Straightforward eliminations

Example 30 ((A12⊕3​A1,B21)(A_{1}^{2}\oplus 3A_{1},B_{2}^{1}) is not a dual pair in E7E_{7}).

8.2. Eliminating and confirming using Dynkin’s and Carter’s tables

Example 31 ((A3,A1~)(A_{3},\widetilde{A_{1}}) is a dual pair in F4F_{4}).

Example 32 ((A1~,B21)(\widetilde{A_{1}},B_{2}^{1}) and (A1~,2​A1~)(\widetilde{A_{1}},2\widetilde{A_{1}}) are not dual pairs in F4F_{4}).

8.3. Eliminating and confirming using dimension of the centralizer

Lemma 33.

Proof.

Example 34 ((A23,A23)(A_{2}^{3},A_{2}^{3}) is not a dual pair in E8E_{8}).

Example 35 ((B31,B41)(B_{3}^{1},B_{4}^{1}) is a dual pair in E8E_{8}).

Example 36 ((A13,A22′′⊕A1)(A_{1}^{3},A_{2}^{2^{\prime\prime}}\oplus A_{1}) is a dual pair in E6E_{6}).

8.4. Eliminating and confirming based on fixed vectors of χ𝔤~\chi_{\widetilde{\mathfrak{g}}}

Remark 37.

Example 38 ((A22′,A13′′⊕A2)(A_{2}^{2^{\prime}},A_{1}^{3^{\prime\prime}}\oplus A_{2}) is a dual pair in E7E_{7}).

9. All of the semisimple dual pairs in the exceptional Lie algebras

9.1. SS-irreducible dual pairs in the exceptional Lie algebras

9.2. Non-SS-irreducible dual pairs in the exceptional Lie algebras

9.2.1. Dual pairs in G2G_{2}

9.2.2. Dual pairs in F4F_{4}

9.2.3. Dual pairs in E6E_{6}

9.2.4. Dual pairs in E7E_{7}

9.2.5. Dual pairs in E8E_{8}

Acknowledgements

References

Appendix A: Subalgebras of type A1A_{1}

Subalgebras of type A1A_{1} in A2A_{2}

Subalgebras of type A1A_{1} in A3A_{3}

2.2. Defining a normalized symmetric form on $\mathfrak{g}$

5. $S$ -irreducible and admissible dual pairs

6.1. Type $A_{n}$

6.2. Type $B_{n}$

6.3. Type $C_{n}$

6.4. Type $D_{n}$

7. Non- $S$ -irreducible dual pairs

Example 30 ( $(A_{1}^{2}\oplus 3A_{1},B_{2}^{1})$ is not a dual pair in $E_{7}$ ).

Example 31 ( $(A_{3},\widetilde{A_{1}})$ is a dual pair in $F_{4}$ ).

Example 32 ( $(\widetilde{A_{1}},B_{2}^{1})$ and $(\widetilde{A_{1}},2\widetilde{A_{1}})$ are not dual pairs in $F_{4}$ ).

Example 34 ( $(A_{2}^{3},A_{2}^{3})$ is not a dual pair in $E_{8}$ ).

Example 35 ( $(B_{3}^{1},B_{4}^{1})$ is a dual pair in $E_{8}$ ).

Example 36 ( $(A_{1}^{3},A_{2}^{2^{\prime\prime}}\oplus A_{1})$ is a dual pair in $E_{6}$ ).

8.4. Eliminating and confirming based on fixed vectors of $\chi_{\widetilde{\mathfrak{g}}}$

Example 38 ( $(A_{2}^{2^{\prime}},A_{1}^{3^{\prime\prime}}\oplus A_{2})$ is a dual pair in $E_{7}$ ).

9.1. $S$ -irreducible dual pairs in the exceptional Lie algebras

9.2. Non- $S$ -irreducible dual pairs in the exceptional Lie algebras

9.2.1. Dual pairs in $G_{2}$

9.2.2. Dual pairs in $F_{4}$

9.2.3. Dual pairs in $E_{6}$

9.2.4. Dual pairs in $E_{7}$

9.2.5. Dual pairs in $E_{8}$

Appendix A: Subalgebras of type $A_{1}$

Subalgebras of type $A_{1}$ in $A_{2}$

Subalgebras of type $A_{1}$ in $A_{3}$

Subalgebras of type $A_{1}$ in $A_{4}$

Subalgebras of type $A_{1}$ in $A_{5}$

Subalgebras of type $A_{1}$ in $A_{6}$

Subalgebras of type $A_{1}$ in $B_{2}$

Subalgebras of type $A_{1}$ in $B_{3}$

Subalgebras of type $A_{1}$ in $B_{4}$

Subalgebras of type $A_{1}$ in $B_{5}$

Subalgebras of type $A_{1}$ in $B_{6}$

Subalgebras of type $A_{1}$ in $C_{2}$

Subalgebras of type $A_{1}$ in $C_{3}$

Subalgebras of type $A_{1}$ in $C_{4}$

Subalgebras of type $A_{1}$ in $C_{5}$

Subalgebras of type $A_{1}$ in $C_{6}$

Subalgebras of type $A_{1}$ in $D_{2}$

Subalgebras of type $A_{1}$ in $D_{3}$

Subalgebras of type $A_{1}$ in $D_{4}$

Subalgebras of type $A_{1}$ in $D_{5}$

Subalgebras of type $A_{1}$ in $D_{6}$

Subalgebras of type $A_{2}$

Subalgebras of type $B_{2}\simeq C_{2}$

Subalgebras of type $G_{2}$

Subalgebras of type $A_{3}\simeq D_{3}$

Subalgebras of type $B_{3}$

Subalgebras of type $C_{3}$