Corollary. For each abelian Lie algebra $\mathfrak{g}$, its universal enveloping algebra is given by $(\mathfrak{S(g)},i)$, where:
$\cdot\mathfrak{S(g)}$ is the symmetric algebra of $\mathfrak{g}$, i.e. $$\mathfrak{S(g)}:=T(\mathfrak{g})/I, $$ where $I \subset T(\mathfrak{g})$ is the two-sided ideal generated by the elements $x \otimes y-y \otimes x$, with $x,y \in \mathfrak{g}$,
$\cdot i:\mathfrak{g} \rightarrow \mathfrak{S(g)}$ is the injective linear map given by $i(x)=x+I.$
I'm try to understand this corollary, but i can't show why the linear map $i:\mathfrak{g} \rightarrow \mathfrak{S(g)}$ is injective.
It should be simple but I can't prove it. Is the anyone who can help me?