  We prove that a closed oriented Einstein four-manifold is either
anti-self-dual or (after passing to a double Riemann cover if necessary)
K\"ahler-Einstein, provided that $\lambda_2 \geq -\frac{S}{12}$, where
$\lambda_2$ is the middle eigenvalue of the self-dual Weyl tensor $W^+$ and $S$
is the scalar curvature. The same conclusion holds for closed oriented
four-manifolds with $\delta W^+=0$.
