  We study rigidity questions for pairs of Lie algebras
$(\mathfrak{g},\mathfrak{n})$ admitting a post-Lie algebra structure. We show
that if $\mathfrak{g}$ is semisimple and $\mathfrak{n}$ is arbitrary, then we
have rigidity in the sense that $\mathfrak{g}$ and $\mathfrak{n}$ must be
isomorphic. The proof uses a result on the decomposition of a Lie algebra
$\mathfrak{g}=\mathfrak{s}_1\dotplus \mathfrak{s}_2$ as the direct vector space
sum of two semisimple subalgebras. We show that $\mathfrak{g}$ must be
semisimple and hence isomorphic to the direct Lie algebra sum
$\mathfrak{g}\cong \mathfrak{s}_1\oplus \mathfrak{s}_2$. This solves some open
existence questions for post-Lie algebra structures on pairs of Lie algebras
$(\mathfrak{g},\mathfrak{n})$. We prove additional existence results for pairs
$(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{g}$ is complete, and for pairs,
where $\mathfrak{g}$ is reductive with $1$-dimensional center and
$\mathfrak{n}$ is solvable or nilpotent.
