Let $G$ be a semisimple algebraic group of type $E_6$, defined over a perfect field $k$ (so $G$ is a group scheme over $k$ and $G_{\bar{k}}$ is a semisimple algebraic group in the usual sense), and let $T$ be a $k$-subtorus of $G$ of rank $6$, so a (not necessarily split) maximal subtorus of $G$.

Does there exist a strictly smaller semisimple subgroup $H$ of $G$ such that $T$ is a subtorus of $H$? What is the type of such $H$?

For example, can you find $H$ of type $A_2\times A_2\times A_2$ (probably too optimistic)?