Purpose of the anchor map, Lie algebroid

Question

Let $X$ be a scheme. A Lie algebroid $L$ on $X$ is a quasi-coherent $\mathcal{O}_X$-module equipped with a morphism of $\mathcal{O}_X$-modules $\sigma:L\to \mathcal{T}_X$ the tangent sheaf, and a Lie bracket $[-,-]:L\otimes_\Bbb C L\to L$, such that:

• The Lie bracket $[-,-]$ commutes with $\sigma$,
• for $a,b\in L, f\in \mathcal{O}_X$, we have $[a,fb]=f[a,b]+\sigma(a)(f)b$.

---

1. a) What does it mean to say $a,b\in L$, does this mean $a,b$ are global sections of $L$?
2. b) What does the bracket mean? Is this also $a\otimes b \mapsto ab-ba$ where $a,b$ are global sections of $L$?
3. What does the so-called anchor map $\sigma$ do, and why is it named this?