We prove that a closed oriented Einstein four-manifold is either anti-self-dual or (after passing to a double Riemannian cover if necessary) Kähler–Einstein, provided that λ2≥-S12, where λ2 is the middle eigenvalue of the self-dual Weyl tensor W+ and S is the scalar curvature. An analogous result holds for closed oriented four-manifolds with δW+=0. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.