If $\mathfrak{g}$ is semisimple and complex, then we can identify a Cartan subalgebra $\mathfrak{h}$ and with choice of positive roots, a Borel subalgebra containing this Cartan.
We can then consider $U(\mathfrak{g})$ and with respect to our choice of Borel sugalgebra $\mathfrak{b}$ above, we can choose a $1$-dimensional $\mathfrak{b}$ representation. By PBW we can quickly understand the Verma modules of $U(\mathfrak{g})$ and thus by an equivalence of categories, consider the representations of $\mathfrak{g}$.
We can consider $U(\mathfrak{g})$ for any Lie algebra, not just those that are semisimple, but without semisimplicity we cannot find a Cartan subalgebra, and hence cannot find a Borel subalgebra to induce a representation from.
Is there an analogue to Verma modules, for reductive Lie algebras or general Lie algebras?