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).