TITLE: Definition of Lie Algebra
QUESTION [9 upvotes]: I have just start studying Lie Algebra. I want to know the motivation for the conditions on the Lie Bracket in the Definition of Lie Algebra. Please explain me or tell me some references. Thanks.

REPLY [5 votes]: Lie algebras arise naturally in multiple contexts. Suppose you have an associative algebra $A$ which is not necessarily commutative. Then a natural thing to do is study how far it deviates from commutativity. That is, it is natural to consider the bracket $[X,Y]=XY-YX$ which measures the failure of commutativity. But now that we've defined this object, we might wonder what properties it satisfies. You can verify through direct computation that this bracket satisfies the Jacobi identity $[X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0$ and also that $[X,X]=0$. So any associative algebra can be made into a Lie algebra.
Another reason the Jacobi identity is natural is that it is an example of an operation satisfying the product rule familiar from differentiation (the Leibniz rule.) You know that the derivative satisfies $D[fg]=(Df)g+f(Dg)$ and any operator $D$ on an algebra satisfying this rule is said to be a derivation. Now one way to write the Jacobi identity is: $[X,[Y,Z]=[[X,Y],Z]+[Y,[X,Z]]$. That is, the function $D_X$ defined by $D_X(Y)=[X,Y]$ satisfies the Leibniz rule when applied to $[Y,Z]$: $D_X[Y,Z]=[D_XY,Z]+[X,D_YZ]$.