Automorphisms of real semisimple Lie algebras and their restricted root systems

Abstract

We prove that every automorphism of the restricted root system of a real semisimple Lie algebra -- when defined properly -- can be lifted to an automorphism of that Lie algebra. In particular, this can be applied to automorphisms of the Dynkin diagram of the restricted root system. We also discuss some applications of this result to the theory of symmetric spaces of noncompact type.