  We show that a basis of a semisimple Lie algebra of compact type, 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 on nilpotent Lie algebras. The result
follows from a more general characterization for diagonalizing the Ricci tensor
for homogeneous spaces. Finally, we also study the Ricci flow behavior of
diagonal metrics on cohomogeneity one manifolds.
