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$. 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). Here the discussion in Borel's book (and mine) implies that the almost-direct product decomposition of $T$ is actually a direct product; but this isn't stated explicitly and doesn't play any role in the underlying theory.

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).