Tate twists of Hodge structures arising from abelian varieties of type IV

We show that certain abelian varieties $A$ have the property that for every Hodge structure $V$ in the cohomology of $A$, every effective Tate twist of $V$ occurs in the cohomology of some abelian variety. We deduce the general Hodge conjecture for certain non-simple abelian varieties of type IV.