Automorphisms of semisimple real Lie algebras (Q919451)

Consider a simple complex Lie algebra \({\mathfrak g}^ c\) with real form \({\mathfrak g}\). Let \(Aut_ e{\mathfrak g}^ c\) be the group of all automorphisms of \({\mathfrak g}^ c\) generated by the elements of the form exp ad x with nilpotent ad x. Let \(Aut_ 0{\mathfrak g}\) be the inverse image of \(Aut_ e{\mathfrak g}^ c\) with respect to the map Aut \({\mathfrak g}\to Aut {\mathfrak g}^ c\), \(g\mapsto g\otimes 1\), and \(Aut_ 0({\mathfrak g},{\mathfrak h})\) be the subgroup of \(Aut_ 0{\mathfrak g}\) preserving the Cartan subalgebra \({\mathfrak h}\subset {\mathfrak g}\). The author deduces a necessary and sufficient condition for certain pairs of \(Aut_ 0({\mathfrak g},{\mathfrak h})\) to be conjugate.