Non-diagonal four-dimensional cohomogeneity-one Einstein metrics in various signatures

Most known four-dimensional cohomogeneity-one Einstein metrics are diagonal in the basis defined by the left-invariant one-forms, though some essentially non-diagonal ones are known. We consider the problem of explicitly seeking non-diagonal Einstein metrics, and we find solutions which in some cases exhaust the possibilities. In particular we construct new examples of neutral signature non–diagonal Bianchi type VIII Einstein metrics with self–dual Weyl tensor.