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)?



Subtori of groups of type E6
This question is precisely answered by [Borel–de Siebenthal theory](https://en.wikipedia.org/wiki/Borel%E2%80%93de_Siebenthal_theory). Ignoring rationality issues (i.e., base changing to an algebraic closure), and fundamental groups, the possible types of $H$ are $E\_6$, $A\_1 + A\_5$, and $A\_2 + A\_2 + A\_2$ (as you hoped, corresponding to removing the root $\alpha\_4$ in Bourbaki's notation).


