A classification of nilpotent compatible Lie algebras

Abstract

Working over an arbitrary field of characteristic different from 2, we extend the Skjelbred-Sund method to compatible Lie algebras and give a full classification of nilpotent compatible Lie algebras up to dimension 4. In case the base field is cubically closed, we find that there are three isomorphism classes and a one-parameter family in dimension 3, and 12 isomorphism classes, 6 one-parameter families and one 2-parameter family in dimension 4.