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 !)