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
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
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