TITLE: structure of maximal tori in semisimple algebraic groups
QUESTION [5 upvotes]: I feel experts might be able to  answer this question immediately.
Let $G$ be a connected $\mathbb Q$-simple and $\mathbb Q$-isotropic  algebraic group. 
Let $S$ be a maximal $\mathbb Q$-split torus of $G$ and
let $T\supset S$ be a maximal torus defined over $\mathbb Q$. Let $T_a$ be the maximal anisotropic subtorus of $T$. Is $T$ a direct product of $S$ and $T_a$?

REPLY [5 votes]: As the example of nfdc23 shows, the answer is generally no.  But maybe it helps to think about this question in a somewhat wider context, where the notions of split and anisotropic tori arise: the study of a connected reductive algebraic group defined over an arbitrary field $k$ (as in the 1965 paper by Borel and Tits).  In the structure theory of such groups, it quickly becomes clear that the nature of $k$-anisotropic groups depends heavily on $k$ (and is not understood for many familiar fields).    Leaving that aside, Borel and Tits got a lot of unified information about the structure of a $k$-isotropic group.    Modulo the knowledge of $k$-anisotropic groups, this leads ultimately to the Tits classification method.  Of course, the special case $k=\mathbb{Q}$ is part of this story, but the main ideas are developed for all $k$.
Note especially that the question raised here never gets answered explicitly in the structure theory.  Indeed, the maximal $k$-anisotropic subtorus $T_a$ here is mentioned but does not play an important role.   The key players include: a (nontrivial!) maximal $k$-split torus $S$ (unique up to $k$-conjugacy), along with its (reductive) centralizer in $G$ (which of course contains $T_a$), a minimal $k$-parabolic subgroup containing $S$, and various data about the associated root systems and Weyl groups. 
What the general theory reveals is the existence of an almost-direct product $ T=T_a\, S$: see for example Borel's 8.15 in GTM 126.   But toward the end of their respective textbooks, Borel (in his expanded second edition) and Springer (in his later framework of $F$-groups) develop a lot of finer detail about classical groups somewhat in the spirit of the answer by nfdc23.   
One extreme, however, is the case of a quasi-split group $G$, in which a minimal $k$-parabolic subgroup is a Borel subgroup (and which is the only type possible for finite or some other special fields).