TITLE: Jacobi identity - intuitive explanation
QUESTION [56 upvotes]: 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?

REPLY [8 votes]: For several reason one is interested in subspaces $L\subseteq End(V)$, with $V$ a vector space, which are closed under commutators —such things show up all over the place. In this situation, the commutator endows $L$ with an antisymmetric multiplication $L\times L\to L$. 
Now suppose you start instead with a vector space $L$ endowed with an antisymmetric multiplication $L\times L\to L$ and you ask yourself: when is $L$ isomorphic to a subspace of some endomorphism algebra of a vector space closed under commutators? 
The answer is, precisely, that

this happens when $L$ satisfies the Jacobi identity,

and in fact, in a sense which can be made precise, the identity is uniquely characterized by this condition.
Likewise, associativity is the precise condition on a vector space $A$ endowed with a multiplication $A\times A\to A$ which ensures that it is isomorphic to a subspace of an endomorphism algebra closed under composition.
Interestingly, such characterizations are not always possible: one famous example is that of Jordan algebras. A subspace $L$ of an endomorphism algebra $End(V)$ is called a special Jordan algebra if it is closed under the operation $$(x,y)\mapsto (x\circ y+y\circ x)/2.$$ It turns out that there is no set of identities which characterizes such things.