The Ricci-flatness that lurks in weight

We introduce two constructions to obtain left-invariant Ricci-flat pseudo-Riemannian metrics on nilpotent Lie groups, one based on gradings, the other on filtrations, both depending on the combinatorics of the set of weights. As an application, we show that every nilpotent Lie algebra of dimension up to $7$ and every nice nilpotent Lie algebra of dimension up to $9$ admit an indefinite Ricci-flat metric.