  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.
