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B Existence and “Uniqueness” of the Aligned Compact Real Form

We prove in this appendix the crucial result that for any real form of a complex semi-simple Lie algebra, one can always find a compact real form aligned with it [93129].

Let 𝔤0 be a specific real form of the semi-simple, complex Lie algebra 𝔤ℂ. Let 𝔠0 be a compact real form of 𝔤ℂ. We may introduce on 𝔤ℂ two conjugations. A first one (denoted by σ) with respect to 𝔤0 and another one (denoted by τ) with respect to the compact real form 𝔠0. The product of these two conjugations constitutes an automorphism λ = στ of ℂ 𝔤. For any automorphism ϕ we have the identity

ad(ϕZ ) = ϕ adZ ϕ −1, (B.1 )
and, as a consequence, the invariance of the Killing form with respect to the automorphisms of the Lie algebra:
′ ′ −1 ′ − 1 ′ B (ϕZ, ϕZ ) = Tr(ad (ϕ Z) ad(ϕ Z )) = Tr(ϕ ad Z ϕ ϕ ad Z ϕ ) = B (Z, Z ). (B.2 )
The automorphism λ = σ τ is symmetric with respect to the Hermitian product B τ defined by B τ(X, Y ) = − B (X, τ(Y )). Indeed (σ τ)−1τ = τ(σ τ) implies that τ ′ τ ′ B (στ [Z ], Z ) = B (Z, στ[Z ]). Thus its square 2 ρ = (σ τ) is positive definite. It can be proved that ρt(t ∈ ℝ ) is a one-parameter group of internal automorphisms of 𝔤0 such that43 ρtτ = τ ρ− t. It follows that
14 − 14 12 − 12 − 12 − 12 14 − 14 ρ τρ σ = ρ τ σ = ρ ρ τ σ = ρ στ = στ ρ = σρ τρ . (B.3 )
In other words, the conjugation σ always commutes with the conjugation 1 1 &tidle;τ = ρ4τ ρ−4, which is the conjugation with respect to the compact real algebra ρ14[𝔠0]. This shows that the compact real form 14 ρ [𝔠0] is aligned with the given real form 𝔤0.

Note also that if there are two Cartan involutions, θ and ′ θ, defined on a real semi-simple Lie algebra, they are conjugated by an internal automorphism. Indeed, as we just mentioned, then an automorphism φ = ((θθ′)2)14 exists, such that θ and ψ = φθ′φ−1 commute. If ψ ⁄= θ, the eigensubspaces of eigenvalues +1 and − 1 of these two involutions are disitnct but, because they commute, a vector X exists, such that θ[X ] = X and ψ[X ] = − X. For this vector we obtain

θ 0 < B ψ(X, X ) = − B (X, θ[X ]) = − B (X, X ), (B.4 ) 0 < B (X, X ) = − B (X, ψ [X ]) = +B (X, X ),
which constitutes a contradiction, and thus implies θ = ψ. An important consequence of this is that any real semi-simple Lie algebra possesses a “unique” Cartan involution44. In the same way, if 𝔤 is a complex semi-simple Lie algebra, the only Cartan involutions of 𝔤ℝ are obtained from the conjugation with respect to a compact real form of 𝔤; all compact real forms being conjugated to each other by internal automorphisms.

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