Negative eigenvalues of the Ricci operator of solvable metric Lie   algebras
  In this paper we get a necessary and sufficient condition for the Ricci operator of a solvable metric Lie algebra to have at least two negative eigenvalues. In particular, this condition implies that the Ricci operator of every non-unimodular solvable metric Lie algebra or every non-abelian nilpotent metric Lie algebra has this property. 
Yu. G. Nikonorov
math.DG