Journal of Lie Theory Volume 10 (2000) 53–80 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag Berlin

A Clifford algebra approach to simple Lie algebras of real rank two, I: the A_{2} case

Paolo Ciatti^{∗}

Communicated by M. Cowling

Abstract. By using Clifford algebraic methods, we classify all real semisimple Lie algebras of type A_{2} . This approach to the classification avoids the need to discuss the real and complex cases separately, and also provides interesting information about the structure of the algebras.

1. Introduction

Since the time of E. Cartan, in the study of Lie algebras, attention has been focused on the involutions of the algebra. It is well known, however, that if 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫 is the Iwasawa decomposition of 𝔤, then the algebra 𝔤 is uniquely determined by the subalgebra 𝔞 ⊕ 𝔫. Thus it should be possible to classify real semisimple Lie algebras by classifying their Iwasawa subalgebras 𝔞 ⊕ 𝔫. This is the first paper of a series devoted to this task.

This approach was followed in two papers by M. Cowling, A. Dooley, A. Korányi, and F. Ricci [3, 4], in which the authors classify real rank one Lie algebras and describe them. These papers rest on the observation of Korányi [8] that the nilpotent subalgebra in the Iwasawa decomposition of a real semisimple Lie algebra of rank one is either Abelian or lies in the class of algebras introduced by A. Kaplan in [6] and called generalized Heisenberg algebras.

The following observation holds for general real semisimple Lie algebras, and is the basis of our approach to studying them. The nilpotent subalgebra 𝔫 is the semi-direct product of a generalized Heisenberg algebra 𝔫_{0} and a nilpotent subalgebra of the algebra which normalises 𝔫_{0} and centralises the centre. This may be seen as a generalization to arbitrary rank of Korányi’s original observation. In this paper we use this fact to classify and describe simple real Lie algebras with root system of type A_{2}, which is the simplest case of a higher rank algebra.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

∗ The research for this article was carried out at the University of New South Wales and at the Politecnico di Torino, with support from the Australian Research Council and from the Italian Istituto Nazionale di Alta Matematica. I wish to thank Professor Michael Cowling of the University of New South Wales and Professor Fulvio Ricci of the Politecnico di Torino for their interest and valuable advice.

As far as we know, this approach to the classification has never been tried before. In spite of its difficulty, our technique, which avoids complexification, seems to be more natural than the classical method for dealing with real algebras. This approach also leads us to discover some interesting properties of 𝔤, and clarifies the role played by Clifford algebras in the structure of real semisimple Lie algebras.

We further observe that both complex and noncomplex Lie algebras are treated simultaneously in our method. The real rank two algebras associated to root systems of type A_{2} are 𝔰𝔩(3, ℝ), 𝔰𝔩(3, ℂ), 𝔰𝔲^{∗}(6), and 𝔢_{6(−26)}, corresponding to the cases where the root spaces have dimensions 1, 2, 4, and 8 (our notation for Lie algebras follows S. Helgason [5, p. 518]).

Here is an outline of the paper. In Section 2, we list some general results on semisimple Lie algebras, focusing attention on restricted roots and on the subalgebra 𝔪. In this section, we quote most standard results without providing the proofs, which may be found in [5].

Section 3 is devoted to generalized Heisenberg Lie algebras. We give the definition and, after some generalities whose proofs may be found in [6], we study the derivations of a generalized Heisenberg algebra in the cases of interest for us.

In Section 4, we choose a set of positive roots Δ_{+} = {α, β, α + β}, and prove that the maximal nilpotent subalgebra 𝔫, given by

𝔫 = 𝔤_{α} ⊕ 𝔤_{β} ⊕ 𝔤_{α+β},

is a generalized Heisenberg algebra, i.e.,

𝔫 = 𝔳 ⊕ 𝔷,

where 𝔷 = 𝔤_{α+β} = ℝ^{d} is the centre and 𝔳 = 𝔤_{α} ⊕ 𝔤_{β} is a module for the Clifford algebra C(d). We use this fact to give an elementary proof that all roots in a root system of type A_{2} have the same multiplicity, d say, and then we establish that d is in {1, 2, 4, 8}. Finally, we show that the decomposition of 𝔳 into the orthogonal direct sum 𝔤_{α} ⊕ 𝔤_{β} corresponds to the usual decomposition of a C(d)-module into the direct orthogonal sum of two C^{+}(d)-modules.

Section 5 is devoted to the study of the subalgebra 𝔪 of a real simple Lie algebra with root system A_{2}. We determine 𝔪 explicitly and describe its action on 𝔫 and θ𝔫 for d in {1, 2, 4, 8}.

In Section 6, we use the results of Sections 4 and 5 to construct a vector space 𝔤_{d}, endowed with a skew-symmetric product [·, ·]_{d} and an involutive [·, ·]_{d}- automorphism θ_{d}, for all d in {1, 2, 4, 8}.

In the two last sections we prove that [·, ·]_{d} is a Lie bracket, showing that it satisfies the Jacobi identity. In particular, in Section 7, we show that (𝔤_{d}, [·, ·]_{d}) is isomorphic to the matrix Lie algebra 𝔰𝔩(3, 𝔽_{d}), where 𝔽_{d} is equal to ℝ, ℂ and ℍ, when d = 1, d = 2 and d = 4 respectively. In Section 8, we consider the case where d = 8. To check the Jacobi identity in 𝔤_{8}, we prove that for any three elements of 𝔤_{8}, there are a vector subspace 𝔤′ containing all three elements and a one-to-one linear mapping λ from 𝔤′ onto 𝔤_{4} such that

λ[X, Y]_{8} = [λX, λY]_{4}.

Since at that stage we have already proved that (𝔤_{4}, [·, ·]_{4}, θ_{4}) is a Lie algebra, it follows that the Jacobi identity holds in (𝔤_{8}, [·, ·]_{8}, θ_{8}).

Finally, up to isomorphism, there is exactly one irreducible generalized Heisenberg algebra with d-dimensional centre, therefore the calculations in Section 5 determine the commutation relations uniquely, and prove that for each d in {1, 2, 4, 8} there is exactly one real semisimple Lie algebra of type A_{2} for which the dimension of each of the root spaces is d.

2. Preliminaries

In this section, we first recall some familiar facts about semisimple Lie algebras, and then prove some results about the subalgebra 𝔪 in this general context. Let 𝔤 be a real semisimple Lie algebra, not necessarily of real rank 2, and let B be the Killing form of 𝔤. Let θ be a Cartan involution of 𝔤, and

𝔤 = 𝔨 ⊕ 𝔭

be the corresponding Cartan decomposition, where

𝔨 = {X ∈ 𝔤 : θX = X} and 𝔭 = {X ∈ 𝔤 : θX = −X}.

Let c be a positive constant, to be fixed later. For X and Y in 𝔤, let

⟨X, Y ⟩ = −c B(X, θY).  (1)

Then ⟨·, ·⟩ is a positive definite inner product. The corresponding norm will be denoted by ∥ · ∥. It follows from the invariance of the Killing form that

⟨[X, Y], Z⟩ = ⟨X, [θY, Z]⟩ ∀X, Y, Z ∈ 𝔤.  (2)

We will often use (2) without comment in the paper.

Fix a maximal abelian subalgebra 𝔞 in 𝔭. A nonzero linear functional α on 𝔞 is said to be a restricted root of (𝔤, 𝔞) if

𝔤_{α} = {X ∈ 𝔤 : [H, X] = α(H)X}

is nontrivial. The subspace 𝔤_{α} is called a root space, and its dimension is called the multiplicity of the root α and denoted by d_{α}. We will denote the set of roots of (𝔤, 𝔞) by Δ, and by 𝔤_{0} the commutant of 𝔞 in 𝔤. By θ-invariance,

𝔤_{0} = (𝔤_{0} ∩ 𝔭) ⊕ (𝔤_{0} ∩ 𝔨) = 𝔞 ⊕ 𝔪,

say. For α and β in Δ ∪ {0}, we have [𝔤_{α}, 𝔤_{β}] ⊆ 𝔤_{α+β}, and in particular, [𝔪, 𝔤_{β}] ⊆ 𝔤_{β}.

For α in Δ, let H_{α} be the unique vector in 𝔞 such that

B(H_{α}, H) = α(H) ∀H ∈ 𝔞.

For α, β in Δ, set

(α | β) = ⟨H_{α}, H_{β}⟩;

this defines an inner product on 𝔞^{∗} which depends on the constant c in (1). It is well known (see [5, p. 94]) that

[X, θX] = −∥X∥^{2} H_{α} ∀X ∈ 𝔤_{α}.  (3)

We fix H in 𝔞 such that α(H) ≠ 0 for all α in Δ, and order the roots by saying that a root α is positive if α(H) > 0. Then

Δ = Δ_{+} ∪ (−Δ_{+}),

where Δ_{+} is the set of positive roots. Let

𝔫 = ∑_{α∈Δ_{+}} 𝔤_{α}; then θ𝔫 = ∑_{α∈Δ_{+}} 𝔤_{−α}.

It is clear that both 𝔫 and θ𝔫 are nilpotent subalgebras. We obtain the decomposition

𝔤 = θ𝔫 ⊕ 𝔤_{0} ⊕ 𝔫 = θ𝔫 ⊕ (𝔞 ⊕ 𝔪) ⊕ 𝔫.

We now examine the structure of 𝔪 more carefully. We emphasize that the results in this section do not rely on the assumption that 𝔤 is of real rank 2. Much of what follows is “known,” but not codified as here (as far as we know).

Proposition 2.1. Suppose that X, X′ ∈ 𝔤_{γ} \ {0} for some root γ and that ⟨X, X′⟩ = 0. Then

[X, θX′] = [X′, θX] ∈ 𝔪.

If in addition 2γ is not a root, then [X, θX′] ≠ 0.

Proof. To prove that [X, θX′] ∈ 𝔪, we first observe that [X, θX′] lies in 𝔤_{0}. Since the decomposition 𝔤_{0} = 𝔞 ⊕ 𝔪 is orthogonal, [X, θX′] lies in 𝔪 if and only if ⟨H, [X, θX′]⟩ = 0 for every H in 𝔞. We find that

⟨H, [X, θX′]⟩ = ⟨[H, θX], θX′⟩ = −γ(H)⟨X, X′⟩ = 0.

Hence, [X, θX′] ∈ 𝔪. In particular, since θ fixes 𝔪, it follows that

[θX, X′] = θ[X, θX′] = [X, θX′].

Finally, if 2γ is not a root, then [X, θX′] ≠ 0, since

[[θX, X′], X] = [[θX, X], X′] = γ([θX, X])X′,

and γ([θX, X]) ≠ 0 by (3).

For any root γ define

𝔪_{γ} = span{[X, θX′] : X, X′ ∈ 𝔤_{γ} and ⟨X, X′⟩ = 0}.

Since [𝔤_{γ}, 𝔤_{−γ}] ⊆ ℝH_{γ} ⊕ 𝔪_{γ}, we have [𝔤_{γ}, 𝔤_{−γ}] ∩ 𝔪 = 𝔪_{γ}.

Lemma 2.2. If γ ∈ Δ, then 𝔪_{γ} is an ideal in 𝔪.

Proof. Since ad is linear, it suffices to show that ad(M)[X, θX′] lies in 𝔪_{γ} for all M in 𝔪 and all orthogonal vectors X and X′ in 𝔤_{γ}. We find that

[M, [X, θX′]] = [X, [M, θX′]] + [θX′, [X, M]] = [X, θ[M, X′]] + [[M, X], θX′].

In this formula, the right hand side lies in 𝔪_{γ} ⊕ 𝔞 and the left hand side lies in 𝔪. So both sides are in 𝔪 ∩ (𝔞 ⊕ 𝔪_{γ}) = 𝔪_{γ}.

Proposition 2.3.

𝔪 = ∑_{γ∈Δ_{+}} 𝔪_{γ}.

Proof. The inclusion ∑_{γ∈Δ_{+}} 𝔪_{γ} ⊆ 𝔪 follows from Proposition 2.1.

For the converse, let

𝔪_{+} = ∑_{γ∈Δ_{+}} 𝔪_{γ},

and

𝔤_{+} = θ𝔫 ⊕ (𝔞 ⊕ 𝔪_{+}) ⊕ 𝔫.

Since each 𝔪_{γ} is an ideal in 𝔪, 𝔪_{+} is an ideal in 𝔪, and 𝔪_{+} ⊕ 𝔞 is an ideal in 𝔪 ⊕ 𝔞. We prove that 𝔤_{+} is an ideal in 𝔤. First, it is a subalgebra, since 𝔫 and θ𝔫 are subalgebras and

[𝔫, θ𝔫] ⊆ 𝔞 ⊕ 𝔪_{+} ⊕ 𝔫 ⊕ θ𝔫,

and since 𝔞 and 𝔪_{+} preserve root spaces under the adjoint action, both being subspaces of 𝔤_{0}. It remains to show that 𝔤_{+} is an ideal. It is already clear that [𝔫 ⊕ 𝔞 ⊕ θ𝔫, 𝔤_{+}] ⊆ 𝔤_{+}. Let M be a generic element of 𝔪. Since ad(𝔪) preserves root spaces and 𝔪_{+} is an ideal in 𝔪, we have by linearity

ad(M)(𝔫 ⊕ 𝔪_{+} ⊕ 𝔞 ⊕ θ𝔫) ⊆ 𝔫 ⊕ 𝔪_{+} ⊕ 𝔞 ⊕ θ𝔫.

Thus 𝔤_{+} is an ideal in 𝔤. Since 𝔤 is simple, 𝔤_{+} = 𝔤 and so 𝔪_{+} = 𝔪.

Proposition 2.4. Suppose that γ is a root and that 2γ is not a root. Suppose also that X_{1}, X_{2}, X_{3}, X_{4} are pairwise orthogonal vectors in 𝔤_{γ}. Then

[[X_{1}, θX_{2}], X_{2}] = γ([X_{2}, θX_{2}])X_{1},  [[X_{1}, θX_{2}], X_{1}] = −γ([X_{1}, θX_{1}])X_{2}, (4)

[[X_{1}, θX_{2}], X_{3}] = 0,  (5)

[[X_{1}, θX_{2}], [X_{3}, θX_{2}]] = γ([θX_{2}, X_{2}])[X_{1}, θX_{3}],  (6)

and

[[X_{1}, θX_{2}], [X_{3}, θX_{4}]] = 0.  (7)

Proof. The formulae (4) are easy consequences of the Jacobi identity and of the fact that 2γ ∉ Δ, since

[[X_{1}, θX_{2}], X_{2}] = [[X_{2}, θX_{2}], X_{1}] + [[X_{1}, X_{2}], θX_{2}] = γ([X_{2}, θX_{2}])X_{1}.

For (5) we have

[[X_{1}, θX_{2}], X_{3}] = [[θX_{1}, X_{2}], X_{3}] = [[θX_{1}, X_{3}], X_{2}] = −[[X_{3}, θX_{1}], X_{2}].

Permuting the indices twice we get

[[X_{1}, θX_{2}], X_{3}] = −[[X_{3}, θX_{1}], X_{2}] = [[X_{2}, θX_{3}], X_{1}] = −[[X_{1}, θX_{2}], X_{3}].

Hence,

[[X_{1}, θX_{2}], X_{3}] = 0.

Now (6) follows by the Jacobi identity using (4) and (5):

[[X_{1}, θX_{2}], [X_{3}, θX_{2}]] = [X_{3}, [[X_{1}, θX_{2}], θX_{2}]] + [θX_{2}, [X_{3}, [X_{1}, θX_{2}]]]

= γ([θX_{2}, X_{2}])[X_{3}, θX_{1}].

Similarly we get (7) from the Jacobi identity and (5).

Definition 2.5. Let {X_{1}, …, X_{d_{γ}}} be an orthonormal basis of 𝔤_{γ}. When i ≠ j, we set

X_{ij} = [X_{i}, θX_{j}].

Corollary 2.6. Suppose that γ is a root and that 2γ is not a root. Then when i ≠ j and k ≠ l, we have

[X_{ij}, X_{kl}] = γ([θX_{1}, X_{1}]) (δ_{jl}X_{ik} + δ_{ik}X_{jl} − δ_{il}X_{jk} − δ_{jk}X_{il}).  (8)

In particular, 𝔪_{γ} is isomorphic to 𝔰𝔬(d_{γ}) and dim 𝔪_{γ} = \frac{1}{2}d_{γ}(d_{γ} − 1).

Proof. By definition

𝔪_{γ} = span{X_{ij} : i < j}.

We know that X_{ij} ≠ 0 by Proposition 2.1. Since

⟨[X_{i}, θX_{j}], [X_{k}, θX_{l}]⟩ = −⟨[[X_{i}, θX_{j}]X_{l}], X_{k}⟩,

it follows from Proposition 2.4 that ⟨X_{ij}, X_{kl}⟩ ≠ 0 if and only if (i, j) = (k, l) or (i, j) = (l, k). Therefore, {X_{ij} : i < j} is an orthogonal basis of 𝔪_{γ}. From Proposition 2.4 we get (8). These commutation relations show that 𝔪_{γ} ≅ 𝔰𝔬(d_{γ}). Hence, dim 𝔪_{γ} = \frac{1}{2}d_{γ}(d_{γ} − 1).

We recall, from the classification of the algebras 𝔰𝔬(d), that 𝔪_{γ} is simple when d_{γ} ≥ 5 or when d_{γ} = 3, and that when d_{γ} = 4,

𝔪_{γ} = 𝔪_{1} ⊕ 𝔪_{2},

where 𝔪_{1} and 𝔪_{2} are ideals isomorphic to 𝔰𝔬(3), and the decomposition is orthogonal.

Corollary 2.7. Suppose that γ is a root and that 2γ is not a root. Then 𝔪_{γ} acts irreducibly on 𝔤_{γ}. Moreover, if d_{γ} ≥ 2 the action is faithful, that is, if M ∈ 𝔪_{γ} and

[M, X] = 0 ∀X ∈ 𝔤_{γ},

then necessarily M = 0.

Proof. This holds since 𝔪_{γ} acts on 𝔤_{γ} as 𝔰𝔬(d_{γ}) acts on ℝ^{d_{γ}}.

Corollary 2.8. Suppose that γ and δ are roots and that 2γ is not a root. If there exist two orthogonal vectors X_{1} and X_{2} in 𝔤_{γ} such that

[X_{1}, θX_{2}] ∈ 𝔪_{δ},

then 𝔪_{γ} ⊆ 𝔪_{δ}.

Proof. The statement is vacuous if d_{γ} = 1 and obvious if d_{γ} = 2. Assume therefore that d_{γ} ≥ 3. Extend {X_{1}, X_{2}} to an orthogonal basis {X_{1}, X_{2}, …, X_{d_{γ}}} of 𝔤_{γ}. Then the set {[X_{i}, θX_{j}] : i ≠ j} spans 𝔪_{γ}. If i ≥ 3, since [X_{1}, θX_{2}] ∈ 𝔪_{δ}, it follows from (6) that

−γ([θX_{2}, X_{2}])[X_{1}, θX_{i}] = [[X_{1}, θX_{2}], [X_{2}, θX_{i}]] ∈ 𝔪_{δ}.

The same argument shows that [X_{i}, θX_{j}] ∈ 𝔪_{δ} whenever 1 ≤ i < j ≤ d_{γ}, proving the statement.

Proposition 2.9. Let γ and δ be roots, and assume that 2γ ∉ Δ. If the action of 𝔪_{δ} on 𝔤_{γ} is trivial then 𝔪_{γ} ∩ 𝔪_{δ} = {0}. Suppose further that d_{γ} ≥ 3. If 𝔪_{γ} ∩ 𝔪_{δ} = {0}, then the action of 𝔪_{δ} on 𝔤_{γ} is trivial.

Proof. Take M in 𝔪_{γ} ∩ 𝔪_{δ}. Since M ∈ 𝔪_{δ}, we have

[M, Y] = 0 ∀Y ∈ 𝔤_{γ}.

However, M lies in 𝔪_{γ} and the action of 𝔪_{γ} on 𝔤_{γ} is faithful by Corollary 2.7. Hence, M = 0.

Assume that d_{γ} ≥ 3 and 𝔪_{γ} ∩ 𝔪_{δ} = {0}. As in Lemma 2.2, the intersection 𝔪_{γ} ∩ 𝔪_{δ} is an ideal in both 𝔪_{γ} and 𝔪_{δ}. Since [𝔪_{γ}, 𝔪_{δ}] is a subset of 𝔪_{γ} ∩ 𝔪_{δ}, it is trivial. In other words, for all M in 𝔪_{δ} the operator ad(M) lies in the commutant of 𝔪_{γ}. But 𝔪_{γ} ≅ 𝔰𝔬(d_{γ}) and any skew-symmetric operator on ℝ^{d_{γ}} which commutes with 𝔰𝔬(d_{γ}) is zero when d_{γ} ≥ 3.

Note that in the statement of Proposition 2.9 we suppose that d_{γ} ≥ 3. It is clear from a study of 𝔰𝔩(3, ℂ) that this hypothesis is necessary.

3. Generalized Heisenberg Lie algebras

In this section, we introduce generalized Heisenberg Lie algebras, and next examine their derivations. We then examine carefully four examples of these algebras, and finally show that the Iwasawa 𝔫 summands of simple Lie algebras of type A_{2} are generalized Heisenberg Lie algebras.

The following definition of generalized Heisenberg (or H-type) Lie algebra is due to Kaplan [6].

Definition 3.1. Let 𝔫 be a two-step real nilpotent Lie algebra, endowed with an inner product ⟨·, ·⟩. Denote by 𝔳 the orthogonal complement of the centre 𝔷 of 𝔫, and for each Z ∈ 𝔷 define a map J_{Z} : 𝔳 → 𝔳 by

⟨J_{Z}X, Y ⟩ = ⟨[X, Y], Z⟩ ∀X, Y ∈ 𝔳.

We say that 𝔫 is a generalized Heisenberg algebra if J_{Z} is orthogonal whenever ∥Z∥ = 1, or equivalently if

∥J_{Z}X∥ = ∥Z∥ ∥X∥ ∀Z ∈ 𝔷 ∀X ∈ 𝔳.

Since the Lie bracket is skew-symmetric, we have

⟨J_{Z}X, Y ⟩ = −⟨X, J_{Z}Y ⟩ ,

for all X and Y in 𝔳. Hence,

J_{Z}^{2} = −∥Z∥^{2}I.

By polarization, we obtain

⟨J_{Z}X, J_{Z′}X⟩ = −⟨Z, Z′⟩∥X∥^{2} I ,

and

J_{Z}J_{Z′} + J_{Z′}J_{Z} = −2⟨Z, Z′⟩I.

In particular the last relation shows that, if d denotes the dimension of 𝔷, then 𝔳 is a module for the Clifford algebra with d generators C(d) (for the definition and properties of Clifford algebras, see, for instance, [9] or [10]).

Take Z in 𝔷. Let Ĵ_{Z} be the extension to 𝔫 of J_{Z} which acts on the centre as minus the reflection in the hyperplane orthogonal to Z. Then Ĵ_{Z} is an automorphism of the Lie algebra 𝔫.

Fix d ≥ 1. We recall that there exists an H-type algebra with a ddimensional centre. We say that two H-type algebras 𝔫_{1} and 𝔫_{2} are isomorphic if there is an orthogonal Lie isomorphism between 𝔫_{1} and 𝔫_{2}. We say that 𝔫 = 𝔳 ⊕ 𝔷 is irreducible if there is no 𝔳′ ⊂ 𝔳 such that 𝔳′ ⊕ 𝔷 is H-type.

Proposition 3.2. Up to isomorphism there is exactly one irreducible H-type algebra with centre of dimension d. We will denote the irreducible generalized Heisenberg algebra with d-dimensional centre by 𝔫_{d} = 𝔳_{d} ⊕ 𝔷_{d}.

We now discuss the derivations of a generalized Heisenberg algebra, following C. Riehm [11]. Let D(𝔫) be the space of skew-symmetric derivations of 𝔫. If D ∈ D(𝔫), then D, being a derivation, maps 𝔷 into itself, and since D is skew-symmetric it also maps 𝔳 into itself.

Let {Z_{1}, …, Z_{d}} be an orthonormal basis of 𝔷. We will set J_{i} = J_{Z_{i}}. For each pair (i, k) with i ≠ k define a linear endomorphism D_{ik}^{0} of 𝔫_{d} by

D_{ik}^{0}V = J_{i}J_{k}V ∀V ∈ 𝔳,

and

D_{ik}^{0}Z_{l} = 2δ_{il}Z_{k} − 2δ_{kl}Z_{i} ∀1 ≤ l ≤ d.

Then set

D_{𝔷}(𝔫) = span{D_{ik}^{0} : i ≠ k}.

It is easy to verify that D_{𝔷}(𝔫) is a subspace of D(𝔫). We see from the definition that the restrictions of the elements of D_{𝔷}(𝔫) to 𝔷 span the algebra of all linear skew-symmetric maps of 𝔷 into itself. Now, since the D_{ik}^{0} ’s are derivations, an easy computation shows that they satisfy the commutation relations of 𝔰𝔬(d). Hence, D_{𝔷}(𝔫) is a Lie algebra isomorphic to 𝔰𝔬(d).

Let D_{0}(𝔫) denote the space of the skew-symmetric derivations of 𝔫 which act trivially on 𝔷.

Proposition 3.3. Take D in D(𝔫). For all Z in 𝔷 we have

D|_{𝔳}J_{Z} = J_{DZ} + J_{Z}D|_{𝔳}.

Furthermore, D(𝔫) = D_{𝔷}(𝔫) ⊕ D_{0}(𝔫), and D_{𝔷} and D_{0} commute.

Proof. Since D is skew-symmetric and a derivation,

⟨DJ_{Z}X, Y ⟩ = −⟨J_{Z}X, DY ⟩
= −⟨Z, [X, DY]⟩
= −⟨Z, D[X, Y]⟩ + ⟨Z, [DX, Y]⟩
= ⟨DZ, [X, Y]⟩ + ⟨J_{Z}DX, Y ⟩
= ⟨J_{DZ}X, Y ⟩ + ⟨J_{Z}DX, Y ⟩,

for all Z in 𝔷 and all X and Y in 𝔳, which proves the first result.

Let D|_{𝔷} be the restriction of a derivation D to 𝔷. Since D|_{𝔷} is a skewsymmetric linear map of 𝔷 into itself, it is an element of 𝔰𝔬(d). Therefore,

D|_{𝔷} = ∑_{i≠k} c_{ik}D_{ik}^{0}|_{𝔷},

where the coefficients c_{ik} = −c_{ki} in ℝ are uniquely determined by D|_{𝔷}. Let

D̃ = D − ∑_{i≠k} c_{ik}D_{ik}^{0}.  (9)

Then D̃ is a skew-symmetric derivation of 𝔫 acting trivially on 𝔷. Further the decomposition (9) is unique and D_{𝔷} and D_{0} commute, since

D_{𝔷}(𝔫) ∩ D_{0}(𝔫) = {0},

as required.

Now we are interested in studying the generalized Heisenberg algebras with centres of dimension of 1, 2, 4, and 8. For the reader’s convenience we recall that C(1) ≅ ℂ, C(2) ≅ ℍ, C(4) ≅ ℍ(2), and C(8) ≅ ℝ(16), where 𝔽(n) denotes the space of n × n matrices with entries in the field 𝔽.

It is well known (see, for instance, [9] or [10]) that any Clifford algebra C(d) splits into the direct sum of its even and odd parts, denoted by C^{+}(d) and C^{−}(d) respectively, and that C^{+}(d) is a Clifford algebra isomorphic to C(d − 1). Corresponding to this decomposition of C(d), when d in {1, 2, 4, 8}, the C(d)- modules 𝔳_{d} split into the sum 𝔳_{d1} ⊕ 𝔳_{d2}, where 𝔳_{d1} and 𝔳_{d2} are irreducible C^{+}(d)- modules. The algebras C^{+}(1) and C^{+}(2), which are isomorphic to ℝ and ℂ have, up to equivalence, only one irreducible module: the irreducible module for C^{+}(1) is isomorphic to ℝ and that for C^{+}(2) is isomorphic to ℂ. However, C^{+}(4) ≅ ℍ ⊕ ℍ and C^{+}(8) ≅ ℝ(8) ⊕ ℝ(8) have two inequivalent irreducible modules, 𝔳_{d1} and 𝔳_{d2} say. As real vector spaces, the irreducible C^{+}(4)-modules are of dimension 4, while the irreducible modules of C^{+}(8) are of dimension 8. These modules may be distinguished by looking at the action of ϵ = J_{1}J_{2} … J_{d}, which acts as the identity on 𝔳_{d1} and as minus the identity on 𝔳_{d2}.

Corollary 3.4. Suppose that d in {1, 2, 4, 8} and that D ∈ D(𝔫). If D|_{𝔷} = 0 and D 𝔳_{d1} ⊆ 𝔳_{d1}, then the restrictions D|_{𝔳_{d1}} and D|_{𝔳_{d2}} of D are related by the formula

D|_{𝔳_{d1}} = −J_{Z}D|_{𝔳_{d2}}J_{Z}  (10)

for all Z in 𝔷 of norm one.

Proof. Since D acts trivially on 𝔷, we get from Proposition 3.3

D|_{𝔳_{d}}J_{Z} = J_{Z}D|_{𝔳_{d}} ∀Z ∈ 𝔷,

which, for Z with norm one, implies

D|_{𝔳_{d2}} = −J_{Z}D|_{𝔳_{d1}}J_{Z},

as required.

We denote by D_{0}^{+}(𝔫_{d}) the subspace of D_{0}(𝔫_{d}) which consists of derivations preserving the C^{+}(d)-modules 𝔳_{d1} and 𝔳_{d2}. In general D_{0}^{+}(𝔫_{d}) is a proper subspace of D_{0}(𝔫_{d}). We observe however that for d in {4, 8}, when 𝔳_{d} is the sum of two inequivalent modules, then necessarily D_{0}^{+}(𝔫_{d}) = D_{0}(𝔫_{d}) since any D in D_{0}^{+}(𝔫_{d}) which commutes with J_{Z} also commutes with ϵ.

Proposition 3.5. The dimensions of the spaces D_{0}^{+}(𝔫_{d}) are equal to 0, 1, 3, and 0 when d is equal to 1, 2, 4, and 8 respectively.

Proof. As we observed 𝔳_{d1} and 𝔳_{d2} are irreducible C^{+}(d)-modules. Moreover, any derivation in D_{0}^{+}(𝔫_{d}) commutes with the action of C^{+}(d). Now for d = 1, since C^{+}(1) ≅ ℝ, the commutant is ℝ. For d = 2, since C^{+}(2) ≅ ℂ, the commutant is ℂ. For d = 4, since C^{+}(4) ≅ ℍ ⊕ ℍ, the commutant is ℍ. For d = 8, since C^{+}(8) ≅ ℝ(8) ⊕ ℝ(8), the commutant is ℝ.

We conclude this section by showing that generalized Heisenberg algebras occur naturally in real simple Lie algebras. This provides the link between the Clifford algebraic approach just presented and semisimple Lie algebras.

Lemma 3.6. Suppose that α, β, α + β ∈ Δ and that 2α + β, α + 2β ∉ Δ. Then

(α | α + β) = (β | α + β) = \frac{1}{2}(α + β | α + β).  (11)

Proof. Recall that if γ, δ are distinct roots, then (see [1, Chapter VI, 1.3])

(γ | δ) < 0 ⇒ γ + δ ∈ Δ,  (12)

moreover, if (γ | δ) = 0, then

γ + δ ∈ Δ ⟺ γ − δ ∈ Δ.  (13)

We also recall that

2\frac{(γ | δ)}{(γ | γ)} ∈ \{0, ±1, ±2, ±3\}.  (14)

We claim that

(α | α + β) > 0 and (β | α + β) > 0.  (15)

Indeed, if (α | α + β) < 0, then 2α + β ∈ Δ by (12), contradicting the hypothesis, and (α | α + β) ≠ 0 by (13) since β ∈ Δ. Thus (α | α + β) > 0; exchanging the roles of α and β shows that (β | α + β) > 0. Since

\frac{(α | α + β)}{(α + β | α + β)} + \frac{(β | α + β)}{(α + β | α + β)} = 1,

(14) and (15) imply (11).

Set

𝔫 = (𝔤_{α} + 𝔤_{β}) ⊕ 𝔤_{α+β},  𝔷 = 𝔤_{α+β}  and 𝔳 = 𝔤_{α} + 𝔤_{β},

so that 𝔫 = 𝔳 ⊕ 𝔷.

Definition 3.7. For Z in 𝔷, we define an operator J_{Z} on 𝔳 by

J_{Z}X = [Z, θX] ∀X ∈ 𝔳.

Theorem 3.8. Fix the constant c in (1) such that

(α + β | α + β) = 2.

Then 𝔫 = 𝔳 ⊕ 𝔷, endowed with the inner product ⟨·, ·⟩, is a generalized Heisenberg algebra with centre 𝔷. In particular, for all Z in 𝔷 and X and Y in 𝔳

⟨J_{Z}X, Y ⟩ = −⟨X, J_{Z}Y ⟩,  (16)
[X, J_{Z}X] = ∥X∥^{2}Z,  (17)

and

⟨J_{Z}X, Y ⟩ = ⟨Z, [X, Y]⟩.  (18)

Proof. We first prove that J_{Z} is skew-symmetric. Take X and Y in 𝔳. From the definition of J_{Z} and (2) it follows that

⟨J_{Z}X, Y ⟩ = ⟨[Z, θX], Y ⟩ = ⟨Z, [X, Y]⟩ = −⟨Z, [Y, X]⟩
= −⟨[Z, θY], X⟩ = −⟨J_{Z}Y, X⟩ = −⟨X, J_{Z}Y ⟩,

i.e., (16) holds.

Now we prove that J_{Z}^{2}X = −∥Z∥^{2}X for all X in 𝔳. Let Z be in 𝔷 and suppose that X is in 𝔤_{α}. From the definition of J_{Z} we get

J_{Z}^{2}X = [Z, θ[Z, θX]].

Hence, from the Jacobi identity

J_{Z}^{2}X = [Z, [θZ, X]] = [[Z, θZ], X] = −α([θZ, Z])X = −∥Z∥^{2}X,  (19)

by (3.8). The case where X ∈ 𝔤_{β} is similar.

It is clear from the structure of the root system that 𝔤_{α+β} ⊆ 𝔷. We prove the converse. It follows from Lemma 3.6 and formula (3.8) that

[X, J_{Z}X] = [X, [Z, θX]] = [Z, [X, θX]] = (α + β)([θX, X])Z

= \frac{1}{2}(α + β | α + β)∥X∥^{2}Z = ∥X∥^{2}Z ∀X ∈ 𝔳.

This proves that for all nonzero X in 𝔳 the vector J_{Z}X is such that [X, J_{Z}X] ≠ 0, whence 𝔳 ∩ 𝔷 = {0}. Therefore, 𝔷 coincides with 𝔤_{α+β}.

4. The structure of 𝔫

From now on, we consider real semisimple Lie algebras with root system A_{2}. This means that

Δ_{+} = {α, β, α + β},  (20)

and that all roots have the same length, hence

(α | β) = −\frac{1}{2}(α | α).  (21)

It is clear that

𝔫 = 𝔤_{α} ⊕ 𝔤_{β} ⊕ 𝔤_{α+β}.

As before, we set

𝔷 = 𝔤_{α+β} and 𝔳 = 𝔤_{α} ⊕ 𝔤_{β}.

If we choose the constant c in (1) so that for all γ in Δ,

⟨H_{γ}, H_{γ}⟩ = 2,

that is

γ([θX, X]) = 2∥X∥^{2},  (22)

for all X in 𝔤_{γ}, then 𝔫, endowed with the inner product ⟨·, ·⟩, is a generalized Heisenberg algebra, by Theorem 3.8. From (21) and (22) it follows that

α([θZ, Z]) = β([θZ, Z]) = ∥Z∥^{2},  (23)

for all Z in 𝔤_{α+β}, and that

α([θY, Y]) = −∥Y ∥^{2} and β([θX, X]) = −∥X∥^{2},  (24)

for all Y in 𝔤_{β} and all X in 𝔤_{α}.

Lemma 4.1. For all nonzero vectors X in 𝔤_{α} the map Z ↦ J_{Z}X from 𝔤_{α+β} into 𝔤_{β} is one-to-one and onto. Its inverse is ∥X∥^{−2} ad_{X}. Similarly, for all nonzero Y in 𝔤_{β} the map Z ↦ J_{Z}Y from 𝔤_{α+β} into 𝔤_{α} is one-to-one and onto and its inverse is ∥Y ∥^{−2} ad_{Y} . In particular, d_{α} = d_{β} = d_{α+β}.

Proof. If J_{Z}X = 0 and X ≠ 0, it follows that Z = 0 from (17). Therefore, Z ↦ J_{Z}X is one-to-one. This map is also onto because given Y in 𝔤_{β}, using the Jacobi identity and the fact that α − β ∉ Δ, we find that

J_{[X,Y]}X = [[X, Y], θX] = [[X, θX], Y] = β([X, θX])Y = ∥X∥^{2} Y.

The inverse of this map is ∥X∥^{−2} ad_{X} by (17).

By symmetry the same is true for Z ↦ J_{Z}Y .

From the lemma we immediately get the following corollaries.

Corollary 4.2. The following relations hold:

[𝔤_{α}, 𝔤_{β}] = 𝔤_{α+β},  [𝔤_{α+β}, 𝔤_{−β}] = 𝔤_{α} and [𝔤_{α+β}, 𝔤_{−α}] = 𝔤_{β}.  (25)

Actually, as we show in [2], formulae (25) hold in all simple real Lie algebras.

Corollary 4.3. Let {Z_{1}, Z_{2}, …, Z_{d}} be an orthonormal basis of 𝔷 and let X be a unit vector in 𝔤_{α}. Then {X_{k} ≡ J_{1}J_{k}X : 1 ≤ k ≤ d} and {Y_{k} ≡ J_{k}X : 1 ≤ k ≤ d} are orthonormal bases of 𝔤_{α} and 𝔤_{β} respectively. Furthermore,

{X_{ik} ≡ [J_{1}J_{i}X, θJ_{1}J_{k}X] : i < k}  and  {Y_{ik} ≡ [J_{i}X, θJ_{k}X] : i < k}

are orthogonal bases of 𝔪_{α} and 𝔪_{β} respectively.

Theorem 4.4. The root spaces 𝔤_{α}, 𝔤_{β}, and 𝔤_{α+β} all have the same dimension, which can be either 1, 2, 4, or 8. Furthermore, 𝔳 is an irreducible C(d)-module.

Proof. All the roots have the same multiplicity by Lemma 4.1. In other words, the number of generators (dim 𝔷 = dim 𝔤_{α+β}) of the Clifford algebra involved in the generalized Heisenberg algebra is half the dimension of the Clifford module 𝔳 (which is dim 𝔤_{α} + dim 𝔤_{β}). There are few Clifford modules with this property. These are the irreducible modules for the Clifford algebras C(1) = ℂ, C(2) = ℍ, C(4) = ℂ(4), and C(8) = ℝ(16), whose real irreducible modules have dimension 2, 4, 8, 16 respectively (see [9, p. 28]).

Proposition 4.5. The root spaces 𝔤_{α} and 𝔤_{β} are irreducible C^{+}(d_{α})-modules. If d_{α} ∈ {4, 8}, then 𝔤_{α} and 𝔤_{β} are inequivalent C^{+}(d_{α})-modules.

Proof. The first part follows from the discussion preceding Corollary 3.4.

Now assume that d_{α} ∈ {4, 8}. Since ϵ = J_{1}J_{2} … J_{d_{α}} commutes with J_{i}J_{k} for all j, k in {1, …, d_{α}}, and since ϵ^{2} = 1, it follows from the irreducibility of the C^{+}(d_{α})-modules 𝔤_{α} and 𝔤_{β} that ϵ is ±I on 𝔤_{α} and 𝔤_{β}. Since ϵ anti-commutes with J_{1}, it follows that if ϵ = I on 𝔤_{α}, then ϵ = −I on 𝔤_{β} and vice versa.

5. The structure of 𝔪

Lemma 5.1.

𝔪 = 𝔪_{α} + 𝔪_{β} = 𝔪_{α} + 𝔪_{α+β} = 𝔪_{β} + 𝔪_{α+β}.

Proof. Proposition 2.3 implies that 𝔪 = 𝔪_{α} + 𝔪_{β} + 𝔪_{α+β}. Thus, to prove that 𝔪 = 𝔪_{α} + 𝔪_{β}, it suffices to show that

𝔪_{γ+δ} ⊆ 𝔪_{γ} + 𝔪_{δ}.

Suppose that M is in 𝔪_{γ+δ}. By Proposition 2.3, we may write

M = ∑_{i=1}^{m}[Z_{i}, θZ′_{i}],

where Z_{i} and Z′_{i} are orthogonal vectors in 𝔤_{γ+δ} for each index i. It is sufficient to prove that if Z, Z′ ∈ 𝔤_{γ+δ} and ⟨Z, Z′⟩ = 0, then [Z, θZ′] ∈ 𝔪_{γ} + 𝔪_{δ}. Since [𝔤_{γ}, 𝔤_{δ}] = 𝔤_{γ+δ} by Corollary 4.2, there are vectors X_{j} and Y_{j} in 𝔤_{γ} and 𝔤_{δ} respectively such that

Z′ = ∑_{1≤j≤n}[X_{j}, Y_{j}],

where n is a positive integer. By the Jacobi identity,

[Z, θZ′] = ∑_{1≤j≤n}[Z, θ[X_{j}, Y_{j}]]

= ∑_{1≤j≤n}[θX_{j}, θ[θZ, Y_{j}]] + ∑_{1≤j≤n}[[Z, θX_{j}], θY_{j}] ∈ (𝔪_{γ} + 𝔪_{δ}) ⊕ 𝔞.

But the left hand side lies in 𝔪, hence [Z, θZ′] belongs to 𝔪_{γ} + 𝔪_{δ}.

The other equalities are proved similarly.

Lemma 5.2. Let {Z_{1}, Z_{2}} be an orthonormal pair in 𝔷, and X be a unit vector in 𝔳. Then

[Z_{1}, θZ_{2}] = [X, θJ_{1}J_{2}X] + [J_{1}X, θJ_{2}X].  (26)

Proof. Since Z_{2} = [X, J_{2}X] by (17), the Jacobi identity implies (26).

Observe that if d_{α} = 1, then 𝔪 = {0}. For d in {2, 4, 8}, we consider the decomposition

𝔪 = 𝔪_{α+β} ⊕ 𝔪_{α+β}^{⊥}.

The elements of 𝔪 are clearly derivations of the generalized Heisenberg algebra 𝔫. If Z_{1}, Z_{2} are in 𝔤_{α+β} and V is in 𝔳, then

[[Z_{1}, θZ_{2}], V] = [[V, θZ_{2}], Z_{1}] = [Z_{1}, θ[Z_{2}, θV]] = J_{1}J_{2}V.  (27)

Now, since 𝔪_{α+β} ≅ 𝔰𝔬(d_{α}) and since the space of restrictions to 𝔤_{α+β} of the elements of D_{𝔷}(𝔫) is isomorphic to 𝔰𝔬(d_{α}), we see that

𝔪_{α+β} = D_{𝔷}(𝔫).

We will see later that 𝔪_{α+β}^{⊥} = D_{0}^{+}(𝔫).

Proposition 5.3. Suppose that d_{α} ≥ 2. If 𝔪_{α+β} ∩ 𝔪_{α} = {0}, then d_{α} = 2. If d_{α} = 2, then 𝔪_{α+β} ∩ 𝔪_{α} = 𝔪_{α+β} ∩ 𝔪_{β} = 𝔪_{β} ∩ 𝔪_{α} = {0} and 𝔪_{α} ≅ 𝔪_{β} ≅ 𝔰𝔬(2); moreover,

𝔪_{α+β} = ℝ{[θX, J_{1}J_{2}X] + [J_{1}X, θJ_{2}X]},
𝔪_{α+β}^{⊥} = ℝ{[θX, J_{1}J_{2}X] − [J_{1}X, θJ_{2}X]},

and

D_{0}^{+}(𝔫_{2}) = 𝔪_{α+β}^{⊥}.

Proof. Assume that 𝔪_{α+β} ∩ 𝔪_{α} = {0}. If d_{α} were greater than or equal to 3, then the action of 𝔪_{α+β} on 𝔤_{α} would be trivial by Proposition 2.9, yielding a contradiction, since (ad[Z_{1}, θZ_{2}])X = J_{1}J_{2}X. Hence, d_{α} = 2.

For the converse, assume that d_{α} = 2, and suppose by contradiction that 𝔪_{α+β} ∩ 𝔪_{α} ≠ {0}. Then we would have

[Z_{1}, θZ_{2}] = c [X, θJ_{1}J_{2}X],  (28)

with c in ℝ. Applying (28) to Z_{1}, we deduce from the Jacobi identity that

2Z_{2} = c [[X, θJ_{1}J_{2}X], Z_{1}] = c [[Z_{1}, θJ_{1}J_{2}X], X] = c [J_{1}^{2}J_{2}X, X] = cZ_{2},

whence c = 2. On the other hand, (28) applied to X gives

J_{1}J_{2}X = 2[[θX, J_{1}J_{2}X], X] = 2α([θX, X])J_{1}J_{2}X,

which implies that (α | α) = 1/2, yielding a contradiction. Thus, the sum of 𝔪_{α} and 𝔪_{β} is direct, and we conclude from Lemma 5.1 that

𝔪 = 𝔪_{α} ⊕ 𝔪_{β}.

From Corollary 2.6, it follows that 𝔪_{α} ≃ 𝔪_{β} ≃ 𝔪_{α+β} ≃ 𝔰𝔬(2).

From (26), we obtain immediately that

𝔪_{α+β} = span{[X, θJ_{1}J_{2}X] + [J_{1}X, θJ_{2}X]},

which implies that

𝔪_{α+β}^{⊥} = ℝ([X, θJ_{1}J_{2}X] − [J_{1}X, θJ_{2}X]).

It is now clear that

𝔪_{α+β}^{⊥} ⊆ D_{0}^{+}(𝔫_{2}).

Since dim D_{0}^{+}(𝔫_{2}) = 1 by Proposition 3.5, it follows that

𝔪_{α+β}^{⊥} = D_{0}^{+}(𝔫_{2}),

as required.

Lemma 5.4. Assume that d ∈ {4, 8}. If {Z_{1}, Z_{2}, Z_{3}, Z_{4}} is an orthonormal system in 𝔤_{α+β} and X is a vector in 𝔤_{α} such that

J_{1}J_{2}J_{3}J_{4}X = X,  (29)

then

Z_{12} + Z_{34} = Y_{12} + Y_{34}, Z_{13} + Z_{42} = Y_{13} + Y_{42},
Z_{14} + Z_{23} = Y_{14} + Y_{23}, Z_{12} − Z_{34} = X_{34} − X_{12},
Z_{13} − Z_{42} = X_{42} − X_{13}, Z_{14} − Z_{23} = X_{23} − X_{14},
Y_{12} − Y_{34} = X_{12} + X_{34}, Y_{13} − Y_{42} = X_{13} + X_{42},
Y_{14} − Y_{23} = X_{14} + X_{23}.

Notice that when d_{α} = 4, it is always possible to reorder any given orthonormal basis of 𝔤_{α+β} in such a way that (29) holds.

Proof. From (26) and (29) we get

[Z_{3}, θZ_{4}] = [X, θJ_{3}J_{4}X] + [J_{3}X, θJ_{4}X] = −[X, θJ_{1}J_{2}X] + [J_{3}X, θJ_{4}X].

Summing this formula and (26) we obtain

[Z_{1}, θZ_{2}] + [Z_{3}, θZ_{4}] = [J_{1}X, θJ_{2}X] + [J_{3}X, θJ_{4}X],

which gives the first equality enunciated. The others may be proved similarly.

Proposition 5.5. Suppose that d_{α} = 4. Then

𝔪 = (𝔪_{α+β} ∩ 𝔪_{α}) ⊕ (𝔪_{α+β} ∩ 𝔪_{β}) ⊕ (𝔪_{α} ∩ 𝔪_{β}),

and

𝔪_{α+β} ∩ 𝔪_{α} ≅ 𝔪_{α+β} ∩ 𝔪_{β} ≅ 𝔪_{α} ∩ 𝔪_{β} ≅ 𝔰𝔬(3).

If {Z_{1}, Z_{2}, Z_{3}, Z_{4}} is an orthonormal basis of 𝔤_{α+β} such that J_{1}J_{2}J_{3}J_{4}X = X, then

𝔪_{α} ∩ 𝔪_{α+β} = span{Z_{12} − Z_{34}, Z_{13} − Z_{42}, Z_{14} − Z_{23}},
𝔪_{β} ∩ 𝔪_{α+β} = span{Z_{12} + Z_{34}, Z_{13} + Z_{42}, Z_{14} + Z_{23}},

and

𝔪_{α} ∩ 𝔪_{β} = span{X_{12} + X_{34}, X_{13} + X_{42}, X_{14} + X_{23}}.

Finally, D_{0}^{+}(𝔫_{4}) = 𝔪_{α} ∩ 𝔪_{β}.

Proof. From Proposition 5.3, it follows that 𝔪_{α+β} ∩ 𝔪_{α} is a nontrivial ideal in 𝔪_{α} and in 𝔪_{α+β}. However, we see from (29) that

[Z_{12} + Z_{34}, X] = 0,

which implies that 𝔪_{α+β} ≠ 𝔪_{α}, by Corollary 2.7. Since

𝔪_{α+β} ≃ 𝔰𝔬(4) ≃ 𝔰𝔬(3) ⊕ 𝔰𝔬(3),

it follows that 𝔪_{α+β} ∩ 𝔪_{α} ≃ 𝔰𝔬(3).

By Lemma 5.4 we have

Z_{12} + Z_{34} = Y_{12} + Y_{34},

so that

Z_{12} + Z_{34} ∈ 𝔪_{α+β} ∩ 𝔪_{β}.

Since 𝔪_{α+β} ∩ 𝔪_{β} is an ideal it follows from Proposition 2.4 that

Z_{13} + Z_{42} ∈ 𝔪_{α+β} ∩ 𝔪_{β} and Z_{14} + Z_{23} ∈ 𝔪_{α+β} ∩ 𝔪_{β}.

Hence,

𝔪_{α+β} ∩ 𝔪_{β} = span{Z_{12} + Z_{34}, Z_{13} + Z_{42}, Z_{14} + Z_{23}}.

The rest of the proposition follows similarly.

Proposition 5.6. Suppose that d_{α} = 8. Then

𝔪 = 𝔪_{α} = 𝔪_{β} = 𝔪_{α+β} ≅ 𝔰𝔬(8).

Proof. By Proposition 5.3, 𝔪_{α+β} ∩ 𝔪_{α} is a nontrivial ideal in 𝔪_{α} and 𝔪_{α+β}. Since 𝔪_{α} ≃ 𝔰𝔬(8), which is simple, it follows that 𝔪_{α+β} ∩ 𝔪_{α} = 𝔪_{α} and similarly that 𝔪_{α+β} ∩ 𝔪_{α} = 𝔪_{α+β}.

Corollary 5.7. Suppose that d_{α} = 8. For all pairs (i, j) with i ≠ j, there are uniquely determined coefficients x_{ij,kl} and y_{ij,kl} in ℝ such that

[X_{i}, θX_{j}] = ∑_{1≤k<l≤8} x_{ij,kl} [Z_{k}, θZ_{l}],  and  [Y_{i}, θY_{j}] = ∑_{1≤k<l≤8} y_{ij,kl} [Z_{k}, θZ_{l}].

Proof. By Proposition 5.6, 𝔪 = 𝔪_{α} = 𝔪_{β} = 𝔪_{α+β} ≅ 𝔰𝔬(8). Therefore, each of the sets {[X_{i}, θX_{j}]}, {[Y_{i}, θY_{j}]} and {[Z_{i}, θZ_{j}]} (where 1 ≤ i < j ≤ 8) are bases of 𝔪.

Let {Z_{1}, …, Z_{8}} be an orthonormal basis of 𝔤_{α+β}.

Lemma 5.8. There is a unit vector X in 𝔤_{α}, such that

J_{1}J_{2}J_{3}J_{4}X = J_{1}J_{2}J_{5}J_{6}X = J_{1}J_{2}J_{7}J_{8}X = J_{1}J_{3}J_{5}J_{7}X = X.

Proof. The operators J_{1}J_{2}J_{3}J_{4}, J_{1}J_{2}J_{5}J_{6}, J_{1}J_{2}J_{7}J_{8}, and J_{1}J_{3}J_{5}J_{7} commute with each other, are symmetric, and have square equal to the identity, so the lemma follows.

We note that J_{1}J_{2}J_{3}J_{4}X = X if and only if
Z_{4} = [X, J_{1}J_{2}J_{3}X].  (30)

We are interested in studying the subalgebra 𝔤′ generated by X, θX, Z_{1}, θZ_{1}, Z_{2}, θZ_{2}, Z_{3}, θZ_{3}, Z_{4} and θZ_{4}. The set of vectors

{Z_{1}, Z_{2}, Z_{3}, Z_{4}, X, J_{1}X, J_{2}X, J_{3}X, J_{4}X, J_{1}J_{2}X, J_{1}J_{3}X, J_{1}J_{4}X}

spans a nilpotent Lie algebra 𝔫′. This subalgebra of 𝔫, endowed with the restriction of ⟨·, ·⟩, is a generalized Heisenberg algebra. Since there is only one irreducible generalized Heisenberg algebra with four dimensional centre, up to equivalence, 𝔫′ is isomorphic to the irreducible generalized Heisenberg algebra 𝔫_{4}. Since J_{1}J_{2}J_{3}J_{4}X = X the commutation relations in 𝔤′ take the same form as in the case d_{α} = 4. Let

M_{1} = X_{12} + X_{34}, M_{2} = X_{13} + X_{42} and M_{3} = X_{14} + X_{23}.

Lemma 5.9. The vectors M_{1}, M_{2} and M_{3} act trivially on span{Z_{1}, Z_{2}, Z_{3}, Z_{4}} and satisfy

M_{1} = X_{12} + X_{34} = Y_{12} − Y_{34} = Z_{58} − Z_{67},
M_{2} = X_{13} + X_{42} = Y_{13} − Y_{42} = Z_{57} − Z_{86},  (31)
M_{3} = X_{14} + X_{23} = Y_{14} − Y_{23} = Z_{56} − Z_{78}.

Proof. Clearly M_{1}, M_{2} and M_{3} act trivially on span{Z_{1}, Z_{2}, Z_{3}, Z_{4}}. Therefore, M_{1}, M_{2}, and M_{3} lie in the linear span of {Z_{ij} : 5 ≤ i < j ≤ 8}, i.e.,

M_{i} = ∑_{5≤j<k≤8} m_{i,jk} Z_{jk}.  (32)

Since the representation of 𝔰𝔬(8) on ℝ^{8} is faithful, by using Proposition 2.4 and Lemma 5.8 to compare the action of (31) and (32) on {X_{1}, …, X_{8}}, we obtain (31).

6. Construction of the algebras

In this section, for any d in {1, 2, 4, 8}, we build a vector space 𝔤_{d}, and then we endow 𝔤_{d} with a skew-symmetric product [· , ·]. In the next two sections we prove that (𝔤_{d}, [·, ·]) is a Lie algebra. Our construction is motivated by the structural analysis of Sections 2 to 5.

Suppose that d ∈ {1, 2, 4, 8}, and let 𝔷_{d} denote ℝ^{d} with its standard inner product ⟨· , ·⟩. We fix an orthonormal basis {Z_{1}, …, Z_{d}} of 𝔷_{d} once and for all. Let C(d) be the Clifford algebra associated to (ℝ^{d}, ⟨· , ·⟩). Take an irreducible module 𝔳_{d} of C(d), and denote by J : C(d) → End(𝔳_{d}) the corresponding representation. The module 𝔳_{d} is naturally endowed with an inner product ⟨· , ·⟩ with respect to which J_{a} is skew-symmetric for all a in C(d). As we observed in Section 5, for d in {1, 2, 4, 8} the dimension of 𝔳_{d} is 2d. By Proposition 3.2, up to equivalence, there is exactly one irreducible generalized Heisenberg algebra (𝔫_{d}, [·, ·], ⟨· , ·⟩) with centre 𝔷_{d}, where

𝔫_{d} = 𝔳_{d} ⊕ 𝔷_{d}.

Since d ∈ {1, 2, 4, 8}, the module 𝔳_{d} splits into the direct orthogonal sum of two irreducible C^{+}(d)-modules 𝔳_{d1} and 𝔳_{d2}. We know that C^{+}(d) is isomorphic to C(d − 1), and that the C^{+}(d)-modules 𝔳_{d1} and 𝔳_{d2} are equivalent when d is 1 or 2 and inequivalent when d is 4 or 8. If d is 4 or 8, then we may and shall assume that ϵ = J_{1} … J_{d} is the identity on 𝔳_{d1} and minus the identity on 𝔳_{d2} (as before we use the notation J_{i} = J_{Z_{i}}). Take X in 𝔳_{d1} such that ∥X∥ = 1, and then

{−X, J_{1}J_{2}X, J_{1}J_{3}X, …, J_{1}J_{d}X} and {J_{1}X, J_{2}X, J_{3}X, …, J_{d}X}

are orthonormal bases of 𝔳_{d1} and 𝔳_{d2} respectively. We set

X_{i} = J_{1}J_{i}X and Y_{i} = J_{i}X.

We now build the algebra 𝔞 ⊕ 𝔪. Let 𝔞 be ℝ^{2} endowed with the canonical inner product. Then let 𝔞^{∗} be the space of real linear forms on 𝔞 with its canonical inner product denoted by (· | ·). We fix two elements α and β in 𝔞^{∗} satisfying the conditions

(α | α) = (β | β) = 2 and (α | β) = −1.  (33)

Once we have α, β and α + β, we introduce the notation

𝔷 = 𝔤_{α+β}, 𝔳_{d1} = 𝔤_{α} and 𝔳_{d2} = 𝔤_{β},

and call 𝔤_{α}, 𝔤_{β}, 𝔤_{α+β} root spaces. We now define the Lie bracket of an element of 𝔞 with an element of 𝔫_{d}.

Definition 6.1. For all H in 𝔞, we set

[H, X + Y + Z] = α(H)X + β(H)Y + (α + β)(H)Z,

for all X in 𝔤_{α}, Y in 𝔤_{β}, and Z in 𝔤_{α+β}.

It is clear that 𝔞 acts on 𝔫_{d} by derivations. As in Section 2, we associate to α, β and α + β the vectors H_{α}, H_{β}, and H_{α+β} in 𝔞.

Definition 6.2. We next set

𝔪_{𝔷}(d) = D_{𝔷}(𝔫_{d}), 𝔪_{0}^{+}(d) = D_{0}^{+}(𝔫_{d}) and 𝔪_{d} = 𝔪_{𝔷}(d) ⊕ 𝔪_{0}^{+}(d),

and let the Lie bracket in 𝔞 ⊕ 𝔪_{d}, which is an algebra of derivations of 𝔫_{d}, be the ordinary commutator.

Note that the sum in 𝔪_{d} = 𝔪_{𝔷}(d) ⊕ 𝔪_{0}^{+}(d) is direct, by Proposition 3.3. In view of Corollary 2.6 and Proposition 3.5, set

𝔪_{𝔷}(1) = 𝔪_{0}^{+}(1) = {0} and 𝔪_{0}^{+}(8) = {0}.

We choose now a specific basis of D_{0}^{+}(𝔫_{d}). In doing this, given a linear map D : 𝔤_{α} → 𝔤_{α}, we will extend it to a map on 𝔤_{β} by

D = −J_{Z}DJ_{Z},

where Z is any unit vector in 𝔷, and then we extend it to be zero on 𝔷. When d = 2, we set

DX_{1} = 3X_{2} and DX_{2} = −3X_{1}.

It is easy to verify that the extended map D on 𝔫_{2}, obtained as described above, is a derivation.

When d = 4, we set

D_{1}X_{k} = 2(δ_{1k}X_{2} − δ_{2k}X_{1} − δ_{3k}X_{4} + δ_{4k}X_{3}),
D_{2}X_{k} = 2(δ_{3k}X_{1} − δ_{1k}X_{3} + δ_{4k}X_{2} − δ_{2k}X_{4}),
D_{3}X_{k} = 2(δ_{1k}X_{4} − δ_{4k}X_{1} + δ_{2k}X_{3} − δ_{3k}X_{2}).

Again it is easy to verify that the extensions to 𝔫_{4} of these maps are derivations, still called D_{1}, D_{2}, and D_{3} respectively.

Lemma 6.3. The subalgebras 𝔪_{𝔷}(d), 𝔪_{0}^{+}(d) and 𝔞 are ideals in 𝔞 ⊕ 𝔪_{d}.

We are now in a position to begin the construction of 𝔤_{d}. We need two copies of 𝔫_{d}, so we take the Cartesian product

𝔫_{d} × {+, −}.

To simplify notation, we set

𝔫_{d+} = 𝔫_{d} × {+} and 𝔫_{d−} = 𝔫_{d} × {−},

and write

𝔫_{d} × {+, −} = 𝔫_{d+} ⊕ 𝔫_{d−}.

We also set

𝔤_{α+β} = 𝔷_{d} × {+}, 𝔤_{−α−β} = 𝔷_{d} × {−},
𝔤_{α} = 𝔳_{d1} × {+}, 𝔤_{−α} = 𝔳_{d2} × {−},
𝔤_{β} = 𝔳_{d1} × {+}, 𝔤_{−β} = 𝔳_{d2} × {−}.

Correspondingly, for V in 𝔫_{d}, we denote by V^{+} the pair (V, +) and by V^{−} the pair (V, −).

Definition 6.4. Let

𝔤_{d} = 𝔫_{d−} ⊕ (𝔞 ⊕ 𝔪_{d}) ⊕ 𝔫_{d+},

and let θ : 𝔤_{d} → 𝔤_{d} be the linear involution defined to be the identity on 𝔪_{d}, minus the identity on 𝔞, and by

θV^{+} = V^{−} and θV^{−} = V^{+} ∀V ∈ 𝔫_{d}.

Clearly θ^{2} is the identity on 𝔤_{d}. Notice also that the image under θ of an orthonormal basis of 𝔫_{d} is an orthonormal basis of 𝔫_{d−}.

Definition 6.5. Let 𝔞 ⊕ 𝔪_{d} act on 𝔫_{d+} by the action defined in Definitions 6.1 and 6.2. For all H in 𝔞, M in 𝔪_{d} and W^{−} in 𝔫_{d−}, we set

[H + M, W^{−}] = θ[θH + θM, θW^{−}] = θ[−H + M, W^{+}].

Then 𝔞 ⊕ 𝔪_{d} acts by derivations not only on 𝔫_{d+}, but also on 𝔫_{d−}. We will denote by 𝔤_{d+} and 𝔤_{d−} the Lie algebras consisting respectively of the semi-direct product of 𝔫_{d+} and 𝔫_{d−} with 𝔞 ⊕ 𝔪_{d}.

We observe that when d ∈ {1, 2}, since there is only one irreducible C^{+}(d)- module, the distinction between 𝔳_{d1} and 𝔳_{d2} is irrelevant in the subsequent construction. When d ∈ {4, 8}, the C^{+}(d)-modules 𝔳_{d1} and 𝔳_{d2} are inequivalent and we are free to identify one of the root spaces {𝔤_{α}, 𝔤_{β}} with either of them. But, after that, in accordance with Proposition 4.5, all the other root spaces are determined. For instance, we may choose 𝔳_{d1} as the root space 𝔤_{α}, then we have to take 𝔤_{β} isomorphic to 𝔳_{d2}, 𝔤_{−α} isomorphic to 𝔳_{d1}, and 𝔤_{−β} isomorphic to 𝔳_{d2}.

It remains to define the Lie bracket of pairs of vectors lying in root spaces 𝔤_{γ} and 𝔤_{−δ} with γ and δ in Δ_{+}. We do this by analogy with the results of Sections 4 and 5.

Definition 6.6. Take Z^{+} in 𝔤_{α+β}, X^{+} in 𝔤_{α}, and Y^{+} ∈ 𝔤_{β}. We set

[Z^{+}, X^{−}] = J_{Z}X^{+}, [Z^{+}, Y^{−}] = J_{Z}Y^{+},
[Y^{+}, X^{−}] = 0, [X^{−}, X^{+}] = ∥X^{+}∥^{2}H_{α},
[Y^{−}, Y^{+}] = ∥Y^{+}∥^{2}H_{β} and [Z^{−}, Z^{+}] = H_{α+β}∥Z∥^{2}.

Suppose also that d ∈ {2, 4, 8}. For all (i, k) with 1 ≤ i < k ≤ d, we set

[Z_{i}^{+}, Z_{k}^{−}] = D_{ik}^{0}.  (34)

Further, if d = 2, we impose the conditions

−[X_{1}^{+}, X_{2}^{−}] + [Y_{1}^{+}, Y_{2}^{−}] = [X, θJ_{1}J_{2}X] + [J_{1}X, θJ_{2}X] = D_{12}^{0}

and

[X_{1}^{+}, X_{2}^{−}] + [Y_{1}^{+}, Y_{2}^{−}] = −[X, θJ_{1}J_{2}X] + [J_{1}X, θJ_{2}X] = D,

by analogy with Proposition 5.3. These relations define the single Lie brackets [X_{i}^{+}, X_{k}^{−}] and [Y_{i}^{+}, Y_{k}^{−}]. If d = 4, we impose the conditions

D_{12}^{0} − D_{34}^{0} = [X_{1}^{+}, X_{2}^{−}] − [X_{3}^{+}, X_{4}^{−}],
D_{13}^{0} − D_{42}^{0} = [X_{1}^{+}, X_{3}^{−}] − [X_{4}^{+}, X_{2}^{−}],  (35)
D_{14}^{0} − D_{23}^{0} = [X_{1}^{+}, X_{4}^{−}] − [X_{2}^{+}, X_{3}^{−}],

and

D_{12}^{0} + D_{34}^{0} = [Y_{1}^{+}, Y_{2}^{−}] + [Y_{3}^{+}, Y_{4}^{−}],
D_{13}^{0} + D_{42}^{0} = [Y_{1}^{+}, Y_{3}^{−}] + [Y_{4}^{+}, Y_{2}^{−}],  (36)
D_{14}^{0} + D_{23}^{0} = [Y_{1}^{+}, Y_{4}^{−}] + [Y_{2}^{+}, Y_{3}^{−}],

by analogy with Lemma 5.4, and then we set

D_{1} = [X_{1}^{+}, X_{2}^{−}] + [X_{3}^{+}, X_{4}^{−}] = [Y_{1}^{+}, Y_{2}^{−}] − [Y_{3}^{+}, Y_{4}^{−}],
D_{2} = [X_{1}^{+}, X_{3}^{−}] + [X_{4}^{+}, X_{2}^{−}] = [Y_{1}^{+}, Y_{3}^{−}] − [Y_{4}^{+}, Y_{2}^{−}],  (37)
D_{3} = [X_{1}^{+}, X_{4}^{−}] + [X_{2}^{+}, X_{3}^{−}] = [Y_{1}^{+}, Y_{4}^{−}] − [Y_{2}^{+}, Y_{3}^{−}].

These relations define the single brackets [X_{i}^{+}, X_{k}^{−}] and [Y_{i}^{+}, Y_{k}^{−}]. Finally, if d = 8, we define, for i ≠ j,

[X_{i}^{+}, X_{j}^{−}] = ∑_{1≤k<l≤8} x_{ij,kl} D_{kl}^{0},

and

[Y_{i}^{+}, Y_{j}^{−}] = ∑_{1≤k<l≤8} y_{ij,kl} D_{kl}^{0},

where x_{ij,kl} and y_{ij,kl} are given by Corollary 5.7. All the remaining brackets are defined by imposing the condition that

[θU, θV] = θ[U, V] ∀U, V ∈ 𝔤_{d}.

In the last two sections we prove that the skew-symmetric product just defined really is a Lie bracket. Then, since there is exactly one irreducible generalized Heisenberg algebra with d-dimensional centre, up to equivalence, by Proposition 3.2, the considerations in Section 5 determine the commutation relations uniquely, proving that for any d in {1, 2, 4, 8}, there is only one real semisimple Lie algebra with d_{α} = d.

7. An explicit model for 𝔤_{d} when d ∈ {1, 2, 4}

In this section we will prove that the vector space 𝔤_{d} endowed with the product [·, ·] is a Lie algebra when d ∈ {1, 2, 4}, by verifying the Jacobi identity.

We denote by 𝔽_{d} the real numbers, the complex numbers and the quaternions, when d = 1, d = 2, and d = 4 respectively. We write z = a_{0}+a_{1}i+a_{2}j+a_{3}k with a_{1}, a_{2}, a_{3}, a_{4} in ℝ for z in 𝔽_{d}, with the understanding that a_{1} = a_{2} = a_{3} = 0 when d = 1 and a_{2} = a_{3} = 0 when d = 2. Correspondingly, we write Im 𝔽_{d} for \{z ∈ 𝔽_{d} : a_{0} = 0\}. If x ∈ 𝔽_{d} we denote by \overline{x} the conjugate of x and by |x| the modulus of x, i.e.,

\overline{x} = a_{0} − a_{1}i − a_{2}j − a_{3}k and |x|^{2} = x\overline{x} = a_{0}^{2} + a_{1}^{2} + a_{2}^{2} + a_{3}^{2}.

Let 𝔰𝔩(3, 𝔽_{d}) be the Lie algebra of 3 × 3 matrices X with entries in 𝔽_{d} such that tr X = 0 when d ∈ {1, 2} and Re tr X = 0 when d = 4. We identify (𝔤_{d}, [·, ·], θ) with 𝔰𝔩(3, 𝔽_{d}), endowed with the matrix commutator, written [·, ·], and with the Cartan involution Θ, given by

ΘX = −X^{∗},

for X in 𝔰𝔩(3, 𝔽_{d}), where X^{∗} = ^{t}\overline{X}.

For x, y and z in 𝔽_{d}, we set

N(x, y, z) = (\begin{matrix}0 & x & z \\ 0 & 0 & y \\ 0 & 0 & 0 \end{matrix}) .

Define

𝔷_{d} = {N(0, 0, z) : z ∈ 𝔽_{d}},
𝔳_{d1} = {N(x, 0, 0) : x ∈ 𝔽_{d}},
𝔳_{d2} = {N(0, y, 0) : y ∈ 𝔽_{d}},
𝔫_{d+} = 𝔳_{d1} ⊕ 𝔳_{d2} ⊕ 𝔷_{d},
𝔫_{d−} = Θ𝔫_{d+}.

We define the inner product ⟨·, ·⟩ on 𝔫_{d+} by

⟨N(x, y, z), N(x′, y′, z′)⟩ = Re(x\overline{x′} + y\overline{y′} + z\overline{z′})

= \frac{1}{2}(x\overline{x′} + y\overline{y′} + z\overline{z′} + x′\overline{x} + y′\overline{y} + z′\overline{z}).

It may be easily verified that 𝔫_{d}, with this inner product, is a generalized Heisenberg algebra and that the operator J is given by

J_{N(0,0,z)}N(x, y, 0) = N(−z\overline{y}, \overline{x}z, 0).

For q_{1}, q_{2}, q_{3} in 𝔽_{d}, we denote by diag(q_{1}, q_{2}, q_{3}) the 3 × 3 diagonal matrix with diagonal entries q_{1}, q_{2}, q_{3} in 𝔽_{d}. Then we define 𝔞 to be the space of real diagonal 3 × 3 matrices of trace zero, and define H_{α}, H_{β}, and H_{α+β} in 𝔞 by

H_{α} = diag(1, −1, 0), H_{β} = (0, 1, −1) and H_{α+β} = (1, 0, −1).

Next, if d = 2, then we define

𝔪_{𝔷} = {diag(ix, 0, −ix) : x ∈ ℝ} and 𝔪_{0}^{+} = {diag(ix, −2ix, ix) : x ∈ ℝ}.

If d = 4, then we define

𝔪_{𝔷} = {diag(q_{1}, 0, q_{2}) : q_{1}, q_{2} ∈ Im 𝔽_{4}} and 𝔪_{0}^{+} = {diag(0, q, 0) : q ∈ Im 𝔽_{4}}.

In both cases we put

𝔪 = 𝔪_{𝔷} ⊕ 𝔪_{0}^{+}.

It is easy to show that the spaces of matrices introduced in this section may be identified with those introduced in Section 6 and that the commutation relations are the same. This proves that 𝔤_{d} is a Lie algebra (i.e., the Jacobi identity holds).

8. The Jacobi identity for (𝔤_{8}, [·, ·]_{8}, θ_{8})

Recall from Definition 6.5 that 𝔤_{8+} and 𝔤_{8−} are the semi-direct products of 𝔞⊕𝔪_{8} with 𝔫_{8+} and 𝔫_{8−} respectively, hence they are Lie algebras. By linearity, to prove that 𝔤_{8} is a Lie algebra, it therefore suffices to check that

[U, [V, W]] + [V, [W, U]] + [W, [U, V]] = 0

(the Jacobi identity) for (U, V, W) in 𝔫_{8+} × 𝔫_{8−} × 𝔤_{8}.

Suppose that (U, V, W) ∈ 𝔫_{8+}×𝔫_{8−}×𝔤_{8}. There are essentially three different cases which have to be taken into account. The first two, which are equivalent, are those in which U, V ∈ 𝔫_{8+} and W ∈ 𝔫_{8−}, or U ∈ 𝔫_{8+} and V, W ∈ 𝔫_{8−}. The last case is that in which U ∈ 𝔫_{8+}, V ∈ 𝔫_{8−}, and W ∈ 𝔪_{8} ⊕ 𝔞. By linearity, in proving the identity for these cases, we may assume that each of U, V , and W lies in a root space or in 𝔪 ⊕ 𝔞. In what follows, we use the term “root vector” to indicate an element of ∪_{γ∈Δ}𝔤_{γ}.

The next results will enable us to avoid direct verification of the Jacobi identity.

Lemma 8.1. Suppose that d = 8 and that γ is a root in Δ_{+}. The vector space

𝔤_{(γ)} = 𝔤_{−γ} ⊕ (ℝH_{γ} ⊕ 𝔪_{8}) ⊕ 𝔤_{γ},

equipped with [·, ·]_{8}, is a real simple Lie algebra of real rank one isomorphic to 𝔰𝔬(1, 9).

Proof. Let {V_{1}, …, V_{8}} be an orthonormal basis of 𝔤_{γ}. Then by (3)

H_{γ} = [θV_{k}, V_{k}],

and further

{[V_{i}, θV_{j}] : 1 ≤ i < j ≤ 8}

is a basis of 𝔪_{8} acting on 𝔤_{γ} and 𝔤_{−γ} by formulae (4) and (5).

Let e_{i} be the row vector whose entry in the i^{th} place is equal to 1 and all other entries are equal to 0. Define

E_{i} = ( \begin{matrix}0 & e_{i} & 0 \\ ^{t}e_{i} & 0 & ^{t}e_{i} \\ 0 & −e_{i} & 0 \end{matrix}) ∀i ∈ \{1, …, 8\}.

Then set

A = (\begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}) .

It is clear that

[E_{i}, ΘE_{j}] = 2 (\begin{matrix}0 & 0 & 0 \\ 0 & e_{ij} & 0 \\ 0 & 0 & 0 \end{matrix}) ∀i, j ∈ \{1, …, 8\} i ≠ j,

where Θ was introduced in Section 7, and e_{ij} is the matrix with (i, j)^{th}-entry equal to 1 and (j, i)^{th} entry equal to −1, and all other entries zero. Then it may be easily seen that the linear map defined by the conditions that V_{i} ↦ E_{i}, θV_{i} ↦ ΘE_{i}, H_{γ} ↦ A, and [V_{i}, θV_{j}] ↦ [E_{i}, ΘE_{j}], is a Lie algebra isomorphism.

Lemma 8.2. Suppose that {Z_{1}, Z_{2}, Z_{3}, Z_{4}} is orthonormal in 𝔤_{α+β} and X is a unit vector in 𝔤_{α}. Suppose further that

J_{1}J_{2}J_{3}J_{4}X = X.  (38)

The smallest [·, ·]_{8}-closed vector subspace of 𝔤_{8} containing Z_{1}, Z_{2}, Z_{3}, Z_{4}, X and θX is denoted by 𝔤^{†}. This subspace, endowed with the restriction [·, ·]′ of [·, ·]_{8}, is a Lie algebra isomorphic to 𝔤_{4}.

Proof. Set

𝔤_{α+β}^{†} = span{Z_{1}, Z_{2}, Z_{3}, Z_{4}},
𝔤_{α}^{†} = span{X, J_{1}J_{2}X, J_{1}J_{3}X, J_{1}J_{4}X},
𝔤_{β}^{†} = span{J_{1}X, J_{2}X, J_{3}X, J_{4}X}

and

𝔫^{†} = 𝔤_{α}^{†} ⊕ 𝔤_{β}^{†} ⊕ 𝔤_{α+β}^{†}.

It is easy to check that (38) ensures that the subspace 𝔫^{†}, equipped with the restrictions of the bracket and of the inner product of 𝔤_{8}, is a subalgebra of the generalized Heisenberg algebra 𝔫_{8+}. Since, up to equivalence, there is only one irreducible generalized Heisenberg algebra with four-dimensional centre, 𝔫^{†} is isomorphic to 𝔫_{4}. Hence the construction of Section 6 yields a Lie algebra 𝔤^{†} isomorphic to 𝔤_{4}, by the results of Section 7. We prove that 𝔤^{†} is a subalgebra of 𝔤_{8}. We observe that for U, V in 𝔤^{†} the definitions of the brackets [U, V]_{4} and [U, V]_{8} are the same with the exception of those brackets which are defined in 𝔤_{4} by (37). However, from Lemma 5.9 we see that [X_{1}, θX_{2}]_{8} + [X_{3}, θX_{4}]_{8} acts on 𝔤_{α}^{†} in exactly the same way as [X_{1}, θX_{2}]_{4} + [X_{3}, θX_{4}]_{4}. Since the action of 𝔪_{α} on 𝔤_{α} is faithful by Lemma 8.1 and Corollary 2.7, this ensures that

[X_{1}, θX_{2}]_{4} + [X_{3}, θX_{4}]_{4} = [X_{1}, θX_{2}]_{8} + [X_{3}, θX_{4}]_{8},

where [·, ·]_{4} stands for the Lie bracket in 𝔤_{4} and [·, ·]_{8} stands for the bracket of 𝔤_{8}. Similarly we deduce that

[X_{1}, θX_{3}]_{4} + [X_{2}, θX_{4}]_{4} = [X_{1}, θX_{3}]_{8} + [X_{2}, θX_{4}]_{8}
[X_{1}, θX_{4}]_{4} + [X_{2}, θX_{3}]_{4} = [X_{1}, θX_{4}]_{8} + [X_{2}, θX_{3}]_{8},

as required.

Proposition 8.3. Take root vectors U, V in 𝔫_{8+} and a root vector W in 𝔫_{8−}. There exists a subspace of 𝔤_{8} containing U, V, W which, equipped with [·, ·]_{8}, is a Lie algebra isomorphic to 𝔤_{4}.

Proof. Given any triple of root vectors in 𝔫_{8+}×𝔫_{8+}×𝔫_{8−} we will use Lemma 5.8 to see that there exist an orthonormal system {Z_{1}, Z_{2}, Z_{3}, Z_{4}} in 𝔤_{α+β} and a unit vector X in 𝔤_{α} such that U, V , and W lie in the span of {Z_{1}, Z_{2}, Z_{3}, Z_{4}}, {−X, J_{1}J_{2}X, J_{1}J_{3}X, J_{1}J_{4}X}, {J_{1}X, J_{2}X, J_{3}X, J_{4}X}, and of the vectors obtained acting on these by means of θ.

Take U and V in 𝔫_{8+} and W in 𝔫_{8−}. We assume without loss of generality that they are unit vectors. There are essentially six different cases to consider.

Case 1. Suppose that U, V ∈ 𝔤_{α+β} and that W ∈ 𝔤_{−α−β}. Consider an orthonormal system {Z_{1}, Z_{2}, Z_{3}} in 𝔤_{α+β} such that

U = Z_{1}, V = v_{1}Z_{1} + v_{2}Z_{2} and θW = w_{1}Z_{1} + w_{2}Z_{2} + w_{3}Z_{3},

where v_{1}, v_{2}, w_{1}, w_{2}, w_{3} ∈ ℝ. We fix a unit vector X in 𝔤_{α}, and define Z_{4} by (30).

Case 2. Suppose that U, V ∈ 𝔤_{α} and that W ∈ 𝔤_{−α}. There is an orthonormal system {Z_{1}, Z_{2}, Z_{3}} in 𝔤_{α+β} such that, writing X for U,

V = v_{1}X + v_{2}J_{1}J_{2}X and θW = w_{1}X + w_{2}J_{1}J_{2}X + w_{3}J_{1}J_{3}X,

where v_{1}, v_{2}, w_{1}, w_{2}, w_{3} ∈ ℝ. We define Z_{4} by (30).

Case 3. Suppose that U ∈ 𝔤_{α+β}, V ∈ 𝔤_{α}, and W ∈ 𝔤_{−α−β}. There exist orthonormal vectors Z_{1} and Z_{2} in 𝔤_{α+β} such that

U = Z_{1} and θW = w_{1}Z_{1} + w_{2}Z_{2},

where w_{1}, w_{2} ∈ ℝ. Setting V = X, we take any unit vector Z_{3} in 𝔤_{α+β} orthogonal to Z_{1} and Z_{2}, and we define Z_{4} by (30).

Case 4. Suppose that U ∈ 𝔤_{α+β}, V ∈ 𝔤_{α} and W ∈ 𝔤_{−α}. There exist orthonormal vectors Z_{1} and Z_{2} in 𝔤_{α+β} such that, setting V = X, then

U = Z_{1} and θW = w_{1}X + w_{2}J_{1}J_{2}X,

where w_{1}, w_{2} ∈ ℝ. We take any unit vector Z_{3} in 𝔤_{α+β} orthogonal to Z_{1} and Z_{2}, and we define Z_{4} by (30).

Case 5. Suppose that U ∈ 𝔤_{α}, V ∈ 𝔤_{β}, and W ∈ 𝔤_{−α}. There exist two orthonormal vectors Z_{1} and Z_{2} in 𝔤_{α+β} such that, if we set U = X, then

V = J_{1}X and θW = w_{1}X + w_{2}J_{1}J_{2}X,

where w_{1}, w_{2} ∈ ℝ. We take any unit vector Z_{3} in 𝔤_{α+β} orthogonal to Z_{1} and Z_{2}, and we define Z_{4} by (30).

Case 6. Suppose that U ∈ 𝔤_{α+β}, V ∈ 𝔤_{β}, and W ∈ 𝔤_{−α}. Setting U = Z_{1}, there exists a unit vector Z_{2} orthogonal to Z_{1} and a unit vector X in 𝔤_{α} such that

V = J_{1}X and W = w_{1}θX + w_{2}θJ_{1}J_{2}X,

where w_{1}, w_{2} ∈ ℝ. We take any unit vector Z_{3} in 𝔤_{α+β} orthogonal to Z_{1} and Z_{2}, and we define Z_{4} by (30).

Since in any case J_{1}J_{2}J_{3}J_{4}X = X by Lemma 5.8, the proposition follows from Lemma 8.2.

Proposition 8.4. Suppose that U is a root vector in 𝔫_{8+}, V is a root vector in 𝔫_{8−}, and L ∈ 𝔞 or L = D_{ij}^{0} . Then there exists a subspace of 𝔤_{8} containing U, V, L which is isomorphic to 𝔤_{4}.

Proof. The case where L ∈ 𝔞 is easy. Indeed, take a unit vector Z in 𝔤_{α+β}. As in the proof of Proposition 8.3, we find a linear subspace of 𝔤_{8} which, endowed with the bracket inherited from 𝔤_{8}, is a Lie algebra isomorphic to 𝔤_{4}. Now, this algebra, being of real rank two, contains 𝔞, and hence also L.

When L ∈ 𝔪_{8}, we may assume that L is equal to D_{12}^{0}. Therefore, there are two orthonormal vectors Z_{1} and Z_{2} in 𝔤_{α+β} such that

L = [Z_{1}, θZ_{2}].

There are essentially three different cases.

Case 1. Suppose that U ∈ 𝔤_{α+β} and that V ∈ 𝔤_{−α−β}. The orthonormal system {Z_{1}, Z_{2}} may be extended to an orthonormal system {Z_{1}, Z_{2}, Z_{3}, Z_{4}} such that

U = u_{1}Z_{1} + u_{2}Z_{2} + u_{3}Z_{3} and V = v_{1}θZ_{1} + v_{2}θZ_{2} + v_{3}θZ_{3} + v_{4}θZ_{4},

where u_{1}, u_{2}, u_{3}, v_{1}, v_{2}, v_{3}, v_{4} ∈ ℝ. Now, since (J_{1}J_{2}J_{3}J_{4})^{2} = I, we may assume that J_{1}J_{2}J_{3}J_{4} has a unit eigenvector X in 𝔤_{α} with eigenvalue 1.

Case 2. Suppose that U ∈ 𝔤_{α} and that V ∈ 𝔤_{−α}. There is an orthonormal system {Z_{1}, Z_{2}, Z_{3}} in 𝔤_{α+β} extending {Z_{1}, Z_{2}}, such that, setting U = X,

V = v_{1}θX + v_{2}θJ_{1}J_{2}X + v_{3}θJ_{1}J_{3}X,

where v_{1}, v_{2}, v_{3} ∈ ℝ. We define Z_{4} by (30).

Case 3. Suppose that U ∈ 𝔤_{α+β} and that V ∈ 𝔤_{−α}. There is an orthonormal system {Z_{1}, Z_{2}, Z_{3}} in 𝔤_{α+β} extending {Z_{1}, Z_{2}}, such that, taking X = θV ,

U = u_{1}Z_{1} + u_{2}Z_{2} + u_{3}Z_{3},

where u_{1}, u_{2}, u_{3} ∈ ℝ. We define Z_{4} by (30).

Since J_{1}J_{2}J_{3}J_{4}X = X in all cases, the result follows from Lemma 8.2.

Theorem 8.5. The vector space 𝔤_{8} endowed with the bracket [·, ·]_{8} defined in Section 6, is a simple real Lie algebra.

Proof. The Jacobi identity holds in 𝔤_{8} by the results above.

We prove that 𝔤_{8} is simple. Suppose that 𝔥 is a nontrivial ideal in 𝔤_{8}, and let U be a nonzero vector in 𝔥. Take Z in 𝔤_{α+β}. Since ad Z is nilpotent, there exists a nonnegative integer k such that

(ad Z)^{k+1}U = 0 and V = (ad Z)^{k}U ≠ 0.

Then V ∈ 𝔥 ∩ 𝔫_{8}, since [Z, V] = 0. Write V as X + Y + Z′, where X ∈ 𝔤_{α}, Y ∈ 𝔤_{β}, and Z′ ∈ 𝔤_{α+β}. Then

[H_{α}, V] = 2X − Y + Z′ ∈ 𝔥, [H_{β}, V] = −X + 2Y + Z′ ∈ 𝔥

and

[H_{α+β}, V] = X + Y + 2Z′ ∈ 𝔥.

From these relations it follows easily that X, Y, Z′ ∈ 𝔥. Assume that X ≠ 0. Since the representation of 𝔪 on 𝔤_{α} is irreducible, it follows that 𝔤_{α} ⊆ 𝔥. Therefore we deduce from Corollary 4.2 that all root spaces are contained in 𝔥, from which it follows that 𝔥 = 𝔤_{8}.

Finally, denoting by B the Killing form of 𝔤_{8}, it is easy to verify that

⟨·, ·⟩ = −B(·, θ·),

as required.

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Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi 24 10129, Torino Italy

Received April 30, 1997 and in final form August 18, 1999