Let $X$ be a (not necessarily smooth or proper) variety over a number field $k$, let be $iota: khookrightarrow mathbb{C}$ an embedding, and let $bar k$ be the algebraic closure of $k$ in $mathbb{C}$. Let $V$ be a subquotient of $H^i(X_{mathbb{C}}^{text{an}}, mathbb{Q})$ such that

- The subquotient $V$ respects the mixed Hodge structure on $H^i(X_{mathbb{C}}^{text{an}}, mathbb{Q})$ (including the $k$-structure coming from the algebraic de Rham cohomology of $X$),
- The subquotient $Votimes mathbb{Q}_ell$ of $H^i(X_{mathbb{C}}^{text{an}}, mathbb{Q})otimes mathbb{Q}_ell=H^i(X_{bar k, text{ét}}, mathbb{Q}_ell)$ respects the Galois action of $text{Gal}(bar k/k)$, and
- $V$ is mixed Tate as a mixed Hodge structure (or $Votimes mathbb{Q}_ell$ is mixed Tate as a Galois representation).

(If you’d like, you can think of $V$ as the realization of a mixed Tate motive over $k$ under any of the various formalisms for mixed Tate motives over number fields — I think that a priori the conditions above are slightly weaker but expected to be equivalent.)

I’d like to know — is the analogue of the Mumford-Tate conjecture is known to be true for such $V$? Explicitly, one expects that the Lie algebra of the image of $$text{Gal}(bar k/k)to GL(Votimes mathbb{Q}_ell)$$

is the extension of scalars to $mathbb{Q}_ell$ of the Lie algebra of a $mathbb{Q}$-group, given as the Tannaka dual of the subcategory of mixed Tate Hodge structures over $k$, generated by $V$.

My feeling is that one can probably extract this from the literature by comparing ranks of various Ext groups (in one’s favorite category of mixed Tate motives and in the $ell$-adic and Hodge settings) but I am hoping it is written explicitly somewhere.