Universal enveloping algebra

For a complex Lie algebra $\mathfrak{g}$, we can define its universal enveloping algebra $U(\mathfrak{g})$ as the complex associative algebra $T(\mathfrak{g})/I$ where $T(\mathfrak{g})$ is the tensor algebra over $\mathfrak{g}$ and $I$ is the ideal generated by $\{ X \otimes Y - Y \otimes X - [X, Y]: X, Y \in \mathfrak{g} \}$. We have the natural map $\iota: \mathfrak{g} \rightarrow U(\mathfrak{g})$. The universal enveloping algebra satisfies the following universal property:

For any complex associative algebra $A$ and linear map $f: \mathfrak{g} \rightarrow A$ satisfying $[f(X), f(Y)] = f(X)f(Y) - f(Y)f(X)$ for all $X, Y \in \mathfrak{g}$, there is a unique algebra homomorphism $f^*: U(\mathfrak{g}) \rightarrow A$ such that $f = f^* \circ \iota$.

Theorem (Poincare-Birkhoff-Witt). Let $\{X_1, \ldots, X_m\}$ be a basis for $\mathfrak{g}$. Then $\{(\iota X_1)^{n_1}\cdots(\iota X_m)^{n_m} : n_1, \ldots, n_m \geq 0\}$ is a basis for $U(\mathfrak{g})$.