I'm asking for a proof or references of the following claim:
Let $V$ be a rational Hodge structure having CM in the sense that its Mumford-Tate group is abelian. Then there is a filtration $F^{\bullet}_{\overline{\mathbb{Q}}}$ on $V_{\overline{\mathbb{Q}}}$ such that the Hodge filtration $F^{\bullet}$ of $V_{\mathbb{C}}$ is given by $$F^{\bullet}=F^{\bullet}_{\overline{\mathbb{Q}}}\otimes_{\overline{\mathbb{Q}}}\mathbb{C}.$$
A CM Hodge structure is a polarizable rational Hodge structure whose Mumford-Tate group is commutative, hence a torus $T$. The Hodge filtration is split by a cocharacter of $T$ over $\mathbb{C}$, which is automatically defined over the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$.
Added: it's the cocharacter $\mu$ such that $\mu(z)$ acts on $V^{p,q}$ as $z^{-p}$. A cocharacter over $\mathbb{C}$ of a torus defined over $\mathbb{Q}^{al}$ is automatically defined over $\mathbb{Q}^{al}$. Since the cocharacter determines the filtration, it also is defined over $\mathbb{Q}^{al}$.
