Kählerity of Einstein four-manifolds

We prove that a closed oriented Einstein four-manifold is either anti-self-dual or (after passing to a double Riemann cover if necessary) Kähler-Einstein, provided that ${\lambda }_2\ge -\frac{S}{12}$, 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 $\delta W^{+}=0$.