  We construct a K3 surface whose transcendental lattice has a self-isomorphism
which is not a linear combination of self-isomorphisms over $\mathbb{Q}$ which
preserve cup products up to nonzero multiples. Products of it with itself give
candidates for counterexamples to the Hodge conjecture which may be of
interest. We also study the Kuga-Satake construction in relation to these
examples.
