  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 $10$
isomorphism classes, $7$ one-parameter families and two $2$-parameter families
in dimension $4$
