Real fundamental Chevalley involutions and conjugacy classes

Abstract

Let $\mathsf{G}$ be a connected reductive linear algebraic group defined over $\mathbb{R}$, and let $C:\mathsf{G}\to \mathsf{G}$ be a fundamental Chevalley involution. We show that for every $g\in \mathsf{G}(\mathbb{R})$, $C(g)$ is conjugate to $g^{-1}$ in the group $\mathsf{G}(\mathbb{R})$. Similar result on the Lie algebras is also obtained.