Let V_1, V_2 be two polarised simple Q-Hodge structures which are non-isomorphic. I am assuming that the Mumford-Tate groups of V_1 and V_2 are semi-simple adjoint.

Is it true in this case that MT(V_1 x V_2) = MT(V_1) x MT(V_2) ?

(I can easily see that this is not true when MT(V_1) and MT(V_2) are tori !)

Let $V_1$, $V_2$ be two polarised simple $Q$-Hodge structures which are non-isomorphic. I am assuming that the Mumford-Tate groups of $V_1$ and $V_2$ are semi-simple adjoint.

Is it true in this case that $MT(V_1 \times V_2) = MT(V_1) \times MT(V_2)$?

(I can easily see that this is not true when $MT(V_1)$ and $MT(V_2)$ are tori !)