I am really struggling with understanding the Jacobi Identity. I am not struggling with verifying it or calculating commutators.. I just can’t see through it. I can’t see the motivation behind it (as an axiom for defining a Lie algebra). Could anyone give an intuitive explanation?
I don’t think there is just one motivation. I will mention three.
First, the Jacobi identity says precisely that the bracket is a derivation with respect to itself, where a derivation of an algebra is a map d with d(a⋅b)=d(a)⋅b+a⋅d(b). Thus, writing ad(a) for the map b↦[a,b], the Jacobi identity may be rewritten (using anti-symmetry) as
Second, given an associative algebra A, one can produce a new, no longer associative, algebra Lie(A) by taking A for the vector space underlying Lie(A), but with new product structure given by (a,b)↦ab−ba. This product, usually written as bracket (commutator) satisfies the Jacobi identity; this is probably the original motivation for the axiom.
Finally, start with a free associative algebra T=T(V) on a vector space V. This turns out (Thm. 1 of §3 “Enveloping algebra of the free Lie algebra”, chapter 1 of Bourbaki´s Lie groups and Lie algebras) to be the enveloping algebra of the free Lie algebra on V: thus just the Jacobi identity is enough to force all the other relations that the sub-Lie algebra of T(V) generated by V with its bracket structure gets. So though one might be tempted to look for other axioms satisfied by commutator, it turns out that the only ones that exist in general are formal consequences of the Jacobi identity and anti-symmetry. This last reason is the most sophisticated, and, at least to my mind, the most convincing demonstration that our axioms are the “right ones”.