Real fundamental Chevalley involutions and conjugacy classes

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