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

Most known four-dimensional cohomogeneity-one Einstein metrics are diagonal in a 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.