If we have two Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ over a field $k$, and if we have a Lie algebra homomorphism $\mathfrak{g}\rightarrow \text{Der}_k(\mathfrak{h})$, then we can define the semi-direct product $\mathfrak{g}\ltimes \mathfrak{h}$: as a $k$-linear space it is just $\mathfrak{g}\oplus\mathfrak{h}$ and the Lie bracket is given by $$ [(g_1,h_1),(g_2,h_2)]=([g_1,g_2],[h_1,h_2]+g_1\cdot h_2-g_2\cdot h_1). $$

Now we have the universal enveloping algebras $U(\mathfrak{g})$, $U(\mathfrak{h})$ and $U(\mathfrak{g}\ltimes \mathfrak{h})$. $\textbf{My question}$ is: could we form a semi-direct product $U(\mathfrak{g})\ltimes U(\mathfrak{h})$ such that $$ U(\mathfrak{g})\ltimes U(\mathfrak{h})\cong U(\mathfrak{g}\ltimes \mathfrak{h})? $$

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