I'm reading Wiebel's Book An Introduction to Homological Algebra, and in the chapter Lie Algebra Homology and Cohomology under Poincare-Birkhoff-Witt theorem there is a Corollary which states without proof that
The map $i : \mathfrak{g} \rightarrow U_{\mathfrak{g}}$ is an injection, so we may identify $\mathfrak{g}$ with $i(\mathfrak{g})$.
I presume that this may be without proof in there because consists a straightforward consequence of the aforementioned theorem, though I don't see why. There might be an easy explanation but I can't "see" it, so can you please help me out?
Also, any kind of reference instead is welcomed!
PBW tells you that for an ordered basis $\{e_\alpha\}$ of $\mathfrak{g}$, the set $\{e_I\}$ where $I$ runs through increasing sequences forms a basis of $U\mathfrak{g}$. In particular, all the $e_I$ are linearly independent. But \begin{align*} i : \mathfrak{g}&\to U\mathfrak{g}\\ e_\alpha&\mapsto e_{(\alpha)}. \end{align*} Thus, $g = \sum_\alpha a_\alpha e_\alpha\in\ker i$ implies $$ 0 = i(g) = \sum_\alpha a_\alpha i(e_\alpha) = \sum_\alpha a_\alpha e_{(\alpha)}, $$ so $a_\alpha = 0$ for all $\alpha$, meaning $g = 0$ and $i$ is injective.