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 $\theta$?