The algebraic and geometric classification of post-$3$-Lie algebras

A post-$3$-Lie algebra is a $3$-Lie algebra with an additional ternary multiplication, skew-symmetric in the first two arguments, related to the bracket by four compatibility identities. We give the algebraic and geometric classification of complex $2$- and $3$-dimensional post-$3$-Lie algebras and of complex $4$-dimensional nilpotent post-$3$-Lie algebras. The variety of $2$-dimensional algebras is irreducible of dimension $4$; the variety of $3$-dimensional algebras has dimension $12$ and $12$ irreducible components, and exactly one rigid algebra; the variety of $4$-dimensional nilpotent algebras has dimension $14$, two irreducible components and no rigid algebras.