Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra, and let $\theta$ be a complex linear involution on $\mathfrak{g}$. Let $\mathfrak{a}$ be a Cartan subspace, and choose a $\theta$-stable Cartan subalgebra $\mathfrak{h}$ containing $\mathfrak{a}$. Finally, make a choice of positive restricted roots in $\mathfrak{a}^\*$, and extend this to a choice of positive roots in $\mathfrak{h}^\*$, giving rise to a positive system for the roots of $\mathfrak{g}$.


Let $\omega$ denote the Chevalley involution of $\mathfrak{g}$ with respect to $\mathfrak{h}$ and the choice of positive system just described. To be explicit, $\omega$ is the unique automorphism of $\mathfrak{g}$ such that $\omega(h)=-h$ for $h\in\mathfrak{h}$, and $\omega(e\_{\pm\alpha})=e\_{\mp\alpha}$ for all simple roots $\alpha$.


Is it true that $\omega$ commutes with $\theta$?



Existence of commuting Chevalley involution
I've realized that the answer is no. I should have checked some examples! The case of $(\mathfrak{gl}(4),\mathfrak{gl}(3)\times\mathfrak{gl}(1))$ gives a counterexample. Indeed, this is induced by the involution $\theta$ given by conjugation by
$$\begin{bmatrix}0 & 0 & 0 & i\\0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\-i &0 &0&0\end{bmatrix}.$$ Then $\theta$ preserves the Cartan subalgebra of diagonal matrices, which contains a Cartan subspace for this pair. One choice of positive system compatible with the Iwasawa decomposition in this case is the usual one, giving the Borel of upper triangular matrices. Thus a Chevalley involution $\omega$ is given by $X\mapsto-X^t$.


Now one can compute that $\omega\theta(e\_{12})=-ie\_{24}$ while $\theta\omega(e\_{12})=ie\_{24}$.


