Let us consider a $R$-Lie algebra ($R$ is a commutative ring) written as a (module) direct sum of two of its subalgebras $$ \mathfrak{g}=\mathfrak{g}_1\oplus\mathfrak{g}_2 $$ and the natural mapping $$ \alpha : \mathcal{U}(\mathfrak{g}_1)\otimes_R\mathcal{U}(\mathfrak{g}_2)\to\mathcal{U}(\mathfrak{g}) $$ ($\mathfrak{g}_i$ are not necessarily ideals).

It is trivial, using PBW theorem that, if the ground ring $R$ is a field, $\alpha$ is one-to-one.

What is true/known in the general case ?

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