I'm asking for a proof or references of the following claim:



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> 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 $$F^{\bullet}=F^{\bullet}\_{\overline{\mathbb{Q}}}\otimes\_{\overline{\mathbb{Q}}}\mathbb{C}.$$
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How to show that Hodge filtration of CM type is algebraic?
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 $\mathbb{Q}^{al}$.


