  We study post-Lie algebra structures on pairs of Lie algebras
$(\mathfrak{g},\mathfrak{n})$, motivated by nil-affine actions of Lie groups.
We prove existence results for such structures depending on the interplay of
the algebraic structures of $\mathfrak{g}$ and $\mathfrak{n}$. We consider the
classes of simple, semisimple, reductive, perfect, solvable, nilpotent, abelian
and unimodular Lie algebras. Furthermore we consider commutative post-Lie
algebra structures on perfect Lie algebras. Using Lie algebra cohomology we
prove that such structures are trivial in several cases. We classify
commutative structures on low-dimensional Lie algebras, and study the case of
nilpotent Lie algebras.
