A Lie algebra is a vector space (over a field; we'll usually use the real numbers or the complex numbers) little g equipped with an operation brackets going from little g times little g to little g satisfying:
bilinearity (over our chosen field k), that is the ability to expand brackets: a X plus b Y brackets Z equals a X bracket Z plus b Y bracket Z, and X bracket (a Y + b Z) equals a X bracket Y plus b X bracket Z, for all X, Y and Z in little g and all a and b in the field k.
for all X in little g we require X bracket X equals 0. This makes sense if you think about the commutator bracket, because X bracket X equals X X minus X X equals 0.
the Jacobi identity: X bracket (Y bracket Z) plus Y bracket (X bracket X) plus Z bracket (X bracket Y) equals zero for all X, Y and Z in little g To remember this formula, note that the second and third terms are cyclic permutations of the first.