  Let $X$ be a smooth projective variety over a finitely generated field $K$ of
characteristic~$0$ and fix an embedding $K \subset \mathbb{C}$. The
Mumford--Tate conjecture is a precise way of saying that certain extra
structure on the $\ell$-adic \'etale cohomology groups of~$X$ (namely, a Galois
representation) and certain extra structure on the singular cohomology groups
of~$X$ (namely, a Hodge structure) convey the same information.
  The main result of this paper says that if $A_1$ and~$A_2$ are abelian
varieties (or abelian motives) over~$K$, and the Mumford--Tate conjecture holds
for both~$A_1$ and~$A_2$, then it holds for $A_1 \times A_2$. These results do
not depend on the embedding $K \subset \CC$.
