On the diagonalization of the Ricci flow on Lie groups

The main purpose of this note is to prove that any basis of a nilpotent Lie algebra for which all diagonal left-invariant metrics have diagonal Ricci tensor necessarily produce quite a simple set of structural constants; namely, the bracket of any pair of elements of the basis must be a multiple of some of them and only the bracket of disjoint pairs can be nonzero multiples of the same element. Some applications to the Ricci flow of left-invariant metrics on Lie groups concerning diagonalization are also given.