  In this paper we study Hodge classes on complex abelian varieties. We prove
some general results that allow us, in certain cases, to compute the Hodge
group of a product abelian variety $X = X_1 \times X_2$ once we know the Hodge
groups of the two factors. Using these results we can compute the Hodge groups
of all abelian varieties of dimension $\leq 5$. We prove that the Hodge ring of
any such abelian variety $X$ is generated by divisor classes together with the
so-called Weil classes on (quotients of) $X$.
