Why isn't Universal enveloping algebra graded?

Question

Given a Lie algebra $L$, define $U(L) = T(L)$ mod $I(L)$ where $T(L)$ is the tensor algebra of $L$ and $I(L)$ is the two sided ideal of $T(L)$ generated by all elements of the form $xy-yx-[x,y]$ where $x,y \in L$. Can somebody explain to me why the generators of $I(L)$ are not homogeneous for the grading of $T(L)$? It seems to me that since every generator is in $L$ this shouldn't be a problem...

Answer 1

You wrote down a generator of $I(L)$: $xy-yx-z$ where $z=[x,y]$. This is inhomogeneous: the $xy$ and $yx$ have degree $2$ but $z\in L$ has degree $1$.