  Let A be an abelian variety defined over a number field K, the number of
torsion points rational over a finite extension L is bounded polynomially in
terms of the degree [L : K]. When A is isogenous to a product of simple abelian
varieties of type I or II in Albert classification and is "fully of Lefschetz
type", i.e. whose Mumford-Tate group is the group of symplectic similitudes
commuting with endomorphisms and which satisfy the Mumford-Tate conjecture, we
compute the optimal exponent for this bound in terms of the dimensions of the
abelian subvarieties of A and their rings of endomorphisms. The result is
unconditional for a product of simple abelian varieties of type I or II with
odd relative dimension. Extending work of Serre, Pink and Hall, we also prove
that the Mumford-Tate conjecture is true for a few new cases for such abelian
varieties.
