Let $A$ and $B$ are Lie subalgebras of a Lie algebra $L$. $U(A)$, $U(B)$ and $U(L)$ are the universal enveloping algebras of $A$, $B$ and $L$, respectively. Let $[A, B]$ be the Lie subalgebras generated by the set {$[a, b]:a\in A, b\in B$} and $U([A, B])$ be he universal enveloping algebras of $[A, B]$. I want to construct $U([A, B])$ in terms of $U(A)$ and $U(B)$. I guess that $U([A, B])$ is the Hopf subalgebra of $U(L)$ generated by the set {$\sum a_{1}b_{1}S(a_{2})S(b_{2}): a\in U(A), b\in U(B)$}, where $S$ is the antipode of the Hopf algebra $U(L)$. Is that so?

`$A, B$`

are nonzero but centralize each other? It's best to work out a number of typical examples in order to get a precise formulation of the question. (Also, does the characteristic of the underlying field matter, or not?) – Jim Humphreys May 1 '13 at 13:08