Let $k$ be a number field. Let $G$ be a quasisplit (but not split) semisimple group over $k$. Let $S$ be a maximal $k$-split torus in $G$. Let $T$ be the centralizer $Z_G(S)$ of $S$; it is a maximal torus in $G$. Does $S$ remain a maximal split torus over nonarchimedean completions $k_v$?
It is certainly not true if $G$ is not necessarily quasisplit, for example if $G$ is an inner form of a split group, like $G=SL_n(D)$.
(This may be obvious but I also don't know much about the splitting behavior of outer forms of split groups.)