  Let $\mathsf G$ be a connected reductive linear algebraic group defined over
$\mathbb R$, and let $C: \mathsf G\rightarrow \mathsf G$ 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.
