  The Cauchy problem for the Zakharov system in the energy-critical dimension
$d=4$ is considered. We prove that global well-posedness holds in the full
(non-radial) energy space for any initial data with energy and wave mass below
the ground state threshold. The result is based on a Strichartz estimate for
the Schr\"odinger equation with a potential. More precisely, a Strichartz
estimate is proved to hold uniformly for any potential solving the free wave
equation with mass below the ground state constraint. The key new ingredient is
a bilinear (adjoint) Fourier restriction estimate for solutions of the
inhomogeneous Schr\"odinger equation with forcing in dual endpoint Strichartz
spaces.
