Lower bounds for ranks of Mumford-Tate groups

Let A be a complex abelian variety and G its Mumford-Tate group. Supposing that the simple abelian subvarieties of A are pairwise non-isogenous, we find a lower bound for the rank rk G of G, which is a little less than log_2 dim A. If we suppose furthermore that End A is commutative, then we can improve this lower bound to rk G≥log_2 dim A+2 and prove that this is sharp. We also obtain the same results for the rank of the ℓ-adic monodromy group of an abelian variety defined over a number field.