  We show that the usual Hodge conjecture implies the general Hodge conjecture
for certain abelian varieties of type III, and use this to deduce the general
Hodge conjecture for all powers of certain 4-dimensional abelian varieties of
type III. We also show the existence of a Hodge structure M such that M occurs
in the cohomology of an abelian variety, but the Tate twist M(1) does not occur
in the cohomology of any abelian variety, even though it is effective.
