I vaguely remember seeing this accurately described somewhere but cannot recall where, so mainly this is a reference request.
Let $T$ be the tangent bundle of a smooth manifold or of an algebraic variety. Commutator of vector fields equips the space of sections of $T$ over each open set with the structure of Lie algebra. I think that the restriction maps are not Lie algebra homomorphisms, since otherwise stalks would be Lie algebras and one would have one canonical $n$-dimensional Lie algebra structure on the tangent space of each $n$-dimensional manifold, which I never heard of.
First question: is the above correct?
Second question: if no, what is this Lie algebra? If yes, what is this structure on the tangent sheaf induced by the commutator of vector fields, does it have an abstract categorical description? Are there examples of similar structures for associative or commutative multiplication?
What I could find: there is a page on nLab about tangent Lie algebroids, with a reference to the paper "Tangent Lie Algebroids" by T. Courant. In principle this should contain a complete answer, but somehow I remember I've seen something more down to earth.
Moreover, for this notion of algebroid I have lots of additional questions:
What is the category of algebroids? What are their morphisms? Can algebroids be equivalently described as some structures on some objects in some category? In other words, does the category of algebroids have a monadic forgetful functor to some simpler category?