Adjoint groups of Mumford-Tate groups

Question

Let $F$ be a sub-field of $\mathbb{C}$ and $B/F$ and $C/F$ be abelian varieties, with $C$ of CM type. Denote the Mumford-Tate groups of $B$, $C$ and $B\times_F C$ by $G_B$, $G_C$ and $G_{B\times C}$, respectively. Does there exist an isomorphism $$ G^{ad}_{B\times C} \cong G^{ad}_B, $$ where $ad$ means the adjoint groups.

I'm looking for a "modern" proof (i.e algebraic groups are seen as group schemes etc...)

Answer 1

Yes, there is obviously a canonical isomorphism. (Since you are characteristic zero, all algebraic group schemes are smooth, so it makes little difference whether you think of them as schemes or varieties.)