  In this paper we investigate the image of the $l$-adic representation
attached to the Tate module of an abelian variety over a number field with
endomorphism algebra of type I or II in the Albert classification. We compute
the image explicitly and verify the classical conjectures of Mumford-Tate,
Hodge, Lang and Tate, for a large family of abelian varieties of type I and II.
In addition, for this family, we prove an analogue of the open image theorem of
Serre.
