While the PoincarĂ©-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is a $\mathbb Q$-algebra. I am wondering if a similar generalization holds for the following fact, which I learnt today from Pavel Etingof:

Let $\mathfrak c$ be a Lie algebra. Let $\mathfrak a$ and $\mathfrak b$ be two Lie subalgebras of $\mathfrak c$ such that $\mathfrak a + \mathfrak b = \mathfrak c$. Clearly, $\mathfrak a \cap \mathfrak b$ is also a Lie subalgebra of $\mathfrak c$. Now, the map

$U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap \mathfrak b\right)} U\left(\mathfrak b\right) \to U\left(\mathfrak c\right),$

$\alpha\otimes_{U\left(\mathfrak a\cap \mathfrak b\right)} \beta\mapsto\alpha\beta$

is an isomorphism (not of algebras, but of $\left(U\left(\mathfrak a\right),U\left(\mathfrak b\right)\right)$-bimodules).

This is proven for free modules using PBW and appropriate bases. I had no time to do any research on this.