Let $L$ be a finite dimensional complex Lie algebra and $U(L)$ be its universal enveloping algebra.
If $A$ is an associateive algebra with $1$ such that
$L$ is a subspace of $A$
$A$ is generated (as algebra) by $L$,
for all $x,y\in L$, the relation $[x,y]=xy-yx$ holds in $A$,
then is it true that $A$ can be embedded in $U(L)$?
I think this is true by applying universal property of $U(L)$.
I am trying to see whether $U(L)$ is the largest (in this sense) associative algebra w.r.t. above three properties.