We define Lie algebras abstractly as algebras whose multiplication satisfies anti-commutativity and Jacobi's Identity. A particular instance of this is an associative algebra equipped with the commutator bracket: $[a,b]=ab-ba$. However, the notation suggests that this bracket is the one we think about. Additionally, the right adjoint to the functor I just mentioned creates the universal enveloping algebra by quotienting the tensor algebra by the tensor version of this bracket; but we could always start with some arbitrary Lie algebra with some other satisfactory bracket and apply this functor.
My question is
"Why the commutator bracket?"
Is it purely from a historical standpoint (and if so could you explain why)? Or is there a result that says any Lie algebra is essentially one with the commutator bracket (maybe something about the faithfulness of the functor from above)?
I know of (a colleague told me) a proof that the Jacobi identity is also an artifact of the right adjoint to the universal enveloping algebra. He can show that it is the necessary identity for the universal enveloping algebra to be associative (if someone knows of this in the literature I would also appreciate the link to this!)
I hope this question is clear, if not, I can revise and try to make it a bit more specific.