  Recently N. Levin (Comp. Math. 127 (2001), 1--21) proved the Tate conjecture
for ordinary cubic fourfolds over finite fields. In this paper we prove the
Tate conjecture for self-products of ordinary cubic fourfolds. Our proof is
based on properties of so called polynomials of K3 type introduced by the
author (Duke Math. J. 72 (1993), 65--83).
