We consider the Cauchy problem for the Zakharov system with a focus on the energy-critical dimension d = 4 and 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 uniform Strichartz estimate for the Schr ̈odinger equation with potentials solving the wave equation. This is joint work with Timothy Candy and Kenji Nakanishi.