Definition of a Lie algebra
As mentioned in the previous chapter, the first examples of Lie algebras were vector spaces of matrices equipped with a ‘multiplication’, which was not the usual matrix multiplication but the commutator [x, y] = xy − yx. The commutator seems to be a fairly defective kind of multiplication, since it is not associative. But, in measuring the failure of associativity, we discover an interesting fact:
Thus we are led to consider algebraic operations [x, y] that satisfy this Jacobi identity, [x, [y, z]] = [[x, y], z] + [y, [x, z]], without necessarily arising as a commutator.
2.1 Definition and first examples