  Every simply connected and connected solvable Lie group $G$ admits a simply
transitive action on a nilpotent Lie group $H$ via affine transformations.
Although the existence is guaranteed, not much is known about which Lie groups
$G$ can act simply transitive on which Lie groups $H$. So far the focus was
mainly on the case where $G$ is also nilpotent, leading to a characterization
depending only on the corresponding Lie algebras and related to the notion of
post-Lie algebra structures.
  This paper studies two different aspects of this problem. First, we give a
method to check whether a given action $\rho: G \to \operatorname{Aff}(H)$ is
simply transitive by looking only at the induced morphism $\varphi:
\mathfrak{g} \to \operatorname{aff}(\mathfrak{h})$ between the corresponding
Lie algebras. Secondly, we show how to check whether a given solvable Lie group
$G$ acts simply transitive on a given nilpotent Lie group $H$, again by
studying properties of the corresponding Lie algebras. The main tool for both
methods is the semisimple splitting of a solvable Lie algebra and its relation
to the algebraic hull, which we also define on the level of Lie algebras. As an
application, we give a full description of the possibilities for simply
transitive actions up to dimension $4$.
