  We refine and generalize the results of K. E. Lauter and E. W. Howe on
principal polarizations on products of abelian varieties over finite fields.
Firstly, we study the reasons for the absence of an irreducible principal
polarization in the isogeny class of the product of an ordinary and a
supersingular abelian variety. Secondly, we provide a necessary condition for
the existence of a principal polarization on an abelian variety in the isogeny
class of the product of a geometrically simple abelian surface and an elliptic
curve. As an application we get that this abelian threefold or its quadratic
twist is a Jacobian.
