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 [93, 129].
Let be a specific real form of the semi-simple, complex Lie algebra . Let be a compact real form of . We may introduce on two conjugations. A first one (denoted by ) with respect to and another one (denoted by ) with respect to the compact real form . The product of these two conjugations constitutes an automorphism of . For any automorphism we have the identity43 . It follows that
Note also that if there are two Cartan involutions, and , defined on a real semi-simple Lie algebra, they are conjugated by an internal automorphism. Indeed, as we just mentioned, then an automorphism exists, such that and commute. If , the eigensubspaces of eigenvalues and of these two involutions are disitnct but, because they commute, a vector exists, such that and . For this vector we obtain44. In the same way, if is a complex semi-simple Lie algebra, the only Cartan involutions of are obtained from the conjugation with respect to a compact real form of ; all compact real forms being conjugated to each other by internal automorphisms.
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