## 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 [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
identity

and, as a consequence, the invariance of the Killing form with respect to the automorphisms of the Lie
algebra:
The automorphism is symmetric with respect to the Hermitian product
defined by . Indeed implies that
. Thus its square is positive definite. It can be
proved that is a one-parameter group of internal automorphisms of such
that
. It follows that
In other words, the conjugation always commutes with the conjugation , which is the
conjugation with respect to the compact real algebra . This shows that the compact real form
is aligned with the given real form .
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 obtain

which constitutes a contradiction, and thus implies . An important consequence
of this is that any real semi-simple Lie algebra possesses a “unique” Cartan
involution.
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.