Diagonalizing the Ricci Tensor

We show that a basis of a semisimple Lie algebra for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being ``nice. Namely, the bracket of any two basis elements is a multiple of another basis element. This extends the work of Lauret and Will cite{lw13} on nilpotent Lie algebras. We also give a characterization for diagonalizing the Ricci tensor for homogeneous spaces, and study the Ricci flow behavior of diagonal metrics on cohomogeneity one manifolds.