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 $$F^{\bullet}=F^{\bullet}_{\overline{\mathbb{Q}}}\otimes_{\overline{\mathbb{Q}}}\mathbb{C}.$$