Diagonalizing cohomogeneity-one Einstein metrics

We investigate when it is possible to globally diagonalize an Einstein metric of cohomogeneity-one under the action of a compact connected Lie group G with isotropy subgroup K. Restricting our attention to the non-monotypic case in which cohomogeneity-one metrics are not automatically diagonalizable, we make use of a subset of the ODEs of the Einstein condition to globally diagonalize the Einstein metric in certain cases. The program we use is inspired by a result well-known to relativists which states that Einstein metrics must be diagonalizable in the Bianchi IX case. We present a full classification of all the cohomogeneity-one metrics on which this method can be used.