What is the point of (Lie) derivations?

Question

Probably some very naive questions, but ...

Definition
Let $A$ be a vector space together with a bilinear map $\mu: A \times A \rightarrow A$. We call a map $D: A \rightarrow A$ a derivation if it satisfies the Leibniz rule $D(\mu (a,b))=\mu(D(a), b) + \mu(a, D(b))$ for all $a,b \in A$. There are many examples of derivations, differential operators on a suitable associative algebra being one of them.

Lie derivations
One can look at derivations of Lie algebras (where $\mu$ is the Lie bracket). Given a Lie algebra $\mathfrak g$ (in fact any vector space with bilinear product) one can show that the derivations on $\mathfrak g$ form a Lie subalgebra of $End(\mathfrak g)$ with respect to the commutator. Furthermore, the Jacobi identity is precisely the statement that $ad_x$ is a derivation with respect to the Lie bracket for any $x \in\mathfrak g$.

Question(s)
Why should we care? What is the relevance of Lie derivations? Why study them? What do they tell us about a Lie algebra? Are they somehow a generalization of differential operators? Are they related to the Lie algebra-Lie group correspondence? Is there any category theoretic perspective on them?

Answer 1

Well, one important application is building the tangent bundle of any finite dimensional manifolds in an intrinsic manner:

Extrinsically, that is assuming our manifold is embedded in some flat space, it turns out that a tangent vector of the manifold can be identified with a 'point' derivation and that their tangent fields can be identified with derivations. Now we can turn all of this around and define tangent vectors and hence tangent bundles and tangent fields for intrinsic manifolds, that is manifolds without an ambient embedding in a flat space.