About enveloping algebras of direct sums This question is imported from MSE. It is linked to this one in the case of semi-direct products. 

My question Let us consider a Lie $R$-algebra ($R$ is a commutative ring) written as a (module) direct sum of two of its subalgebras
  $$
\mathfrak{g}=\mathfrak{g}_1\oplus\mathfrak{g}_2\ (\oplus=\oplus_{R-mod})
$$ 
  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).
On can check, using generators, that $\alpha$ is onto (and, in some usual cases - see below - one-to-one). 
What is true/known in the general case ? 

I put here the explicit construction in case one of the $\mathfrak{g}_i$ is an ideal. The proof goes as follows :
Take it that $\mathfrak{g}_1$ is such. 


*

* Consider the action $\delta : \mathfrak{g}_2\to \mathfrak{Der}
(\mathfrak{g}_1)$ by derivations (adjoint representation)

* Extend $\delta$ to $\mathfrak{Der}(\mathcal{U}(\mathfrak{g}_1))$ as in Bourbaki Lie ch 1 paragraph 2.8 prop 7.

* Extend $\delta$ as a morphism of $R$-algebras $\mathcal{U}(\mathfrak{g}_2)\to\mathrm{End}(\mathcal{U}(\mathfrak{g}_1))$ by universal property 

* Set a law of $R$-unital associative algebra on $\mathcal{U}(\mathfrak{g}_1)\otimes_R\mathcal{U}(\mathfrak{g}_2)$ by 
$$
(u_1\otimes u_2).(v_1\otimes v_2)=(u_1\otimes 1)\Big((\delta\otimes\gamma_2)\circ\Delta(u_2)[v_1]\otimes v_2\Big) 
$$
where $\gamma_2(m)$ is the multiplication by $m$ on the left within $\mathcal{U}(\mathfrak{g}_2)$.
 
 A: I'm adding my comment as a partial answer, as discussed there; this is a reformulation of point 2 that has been added to the question.
Let $\mathcal L$ be the category of Lie $R$-algebras. Assume that $\mathfrak g = \mathfrak g_1 \oplus \mathfrak g_2$ in $\mathcal L$ (direct sum, i.e. I suppose that they commute). Let $\mathcal C$ be the category of unital $R$-algebras. Then $\alpha$ is an isomorphism in $\mathcal C$, since for every unital $R$-algebra $A$, the map
$$\text{Hom}_{\mathcal L}(\mathfrak g,A) \cong \text{Hom}_{\mathcal C}(U(\mathfrak g),A) \stackrel{\alpha^*}{\to} \text{Hom}_{\mathcal C}(U(\mathfrak g_1) \otimes_R U(\mathfrak g_2),A)$$
is bijective (by the Yoneda lemma). This is because the right-hand side is given by all
$$(f_1,f_2) \in \text{Hom}_{\mathcal C}(U(\mathfrak g_1),A) \times \text{Hom}_{\mathcal C}(U(\mathfrak g_2),A) \mbox{ with } [f_1(x_1),f_2(x_2)]=0 \mbox{ for } x_i \in \mathfrak g_i,$$
that is, all pairs of maps $(f_1,f_2)\!: \mathfrak g_1 \oplus \mathfrak g_2 \to A$ in $\mathcal L$ with $[f_1(x_1),f_2(x_2)]=0$ for all $x_i \in \mathfrak g_i$. Then the inverse of $\alpha^*$ is defined by $(f_1,f_2) \mapsto f_1+f_2$, which is well-defined, since
$$(f_1+f_2)([x_1+x_2,y_1+y_2]) = f_1([x_1,y_1]) + f_2([x_2,y_2]) = [f_1(x_1) + f_2(x_2), f_1(y_1) + f_2(y_2)].$$
A: There are probably not many textbook references for these generalities, but I'd suggest looking at the first chapter of the Bourbaki treatise Groupes et algebres de Lie (whose chapters I-IX have been translated into English, possibly from later editions than the 1960 Hermann edition of Chapter I which is at hand).       
Keep in mind that the first three sections of this chapter treat very general Lie algebras over a commutative ring (with 1) called $K$.    The universal enveloping algebra of a "product" of two Lie algebras over $K$ is considered in $\S2.2$.     I think this is essentially the same set-up you start with, in which the two Lie subalgebras commute with each other; otherwise I'm not sure what you mean by "direct sum".    Anyway, in this framework it's fairly easy to prove that the universal enveloping algebra of such a product is isomorphic to the product of the two universal enveloping algebras: see their Proposition 2.   
Are you asking a more subtle question here?  Semi-direct products are of course more difficult to deal with.  
