Let $L$ be a Lie algebra and $U(L)$ be the corresponding universal enveloping algebra. If $L$ is finite-dimensional then by the virtue of PBW theorem we know that $L$ is embedded in $U(L).$ Can we conclude the same thing for infinite dimensional Lie algebra as well? In order to figure it out I started to look at the construction of $U(L)$ which is obtained by quotienting the tensor algebra $T(L)$ modulo the two sided ideal $J$ of $T(L)$ generated by the elements of the form $[x,y] - (xy - yx),$ $x,y \in L.$ For finite-dimensional case the embedding is precisely given by the quotient map $x \mapsto x + J.$ But I firmly believe that it continues to be an embedding for infinite dimensional case as well. For that what we nee to argue is that $L \cap J = (0)$ But I couldn't quite able to prove it. Could anyone please help me in this regard?
Thanks in advance.