Definition: Let $\mathfrak{g}$ be a Lie algebra. A universal enveloping algebra of $\mathfrak{g}$ is a pair $(\mathfrak{U},i)$ consisting of a unital associative algebra $\mathfrak{U}$ and a Lie morphism $i: \mathfrak{g} \rightarrow \mathfrak{U}_L$ satisfying the following universal property:
for every pair $(\sigma, \mathcal{A})$ consisting of a unital associative algebra $\mathcal{A}$ and a Lie algebra morphism $\sigma: \mathfrak{g} \rightarrow \mathcal{A}_L$ there exists a unique unital algebra morphism $\tilde{\sigma}: \mathfrak{U} \rightarrow \mathcal{A}$ such that $\tilde{\sigma} \circ i= \sigma$.
In this definition appears $\mathcal{A}_L$ that is the associated Lie algebra of the unital associative algebra $\mathcal{A}$, but i don't understand how $\mathcal{A}_L$ is constructed from $\mathcal{A}$. Is there anyone who can explain it to me?
