Lie bracket (commutator)

The Lie bracket measures how much two transformations fail to commute. Do A then B, versus B then A; the discrepancy is captured by [A, B] = AB - BA. If the two operations commute, the bracket is zero; the bigger the bracket, the more the order matters. This single number-valued-matrix is the heartbeat of a Lie algebra.

Formally [A, B] = AB - BA is the commutator of two matrices. It is bilinear (linear in each slot), antisymmetric ([A,B] = -[B,A], so [A,A] = 0), and satisfies the Jacobi identity [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0. Those three properties ARE the abstract definition of a Lie algebra's bracket — the commutator is the concrete model.

The bracket is the multiplication of the Lie algebra, replacing the group's matrix product. Crucially, if X and Y lie in a Lie algebra g, so does [X, Y]: the commutator of two skew-symmetric matrices is skew-symmetric, of two anti-Hermitian matrices is anti-Hermitian, of two traceless matrices is traceless (since trace(XY) = trace(YX)). Closure under the bracket is what makes g an algebra, not just a vector space.

Why it matters: the bracket is the FIRST-ORDER record of group non-commutativity. The Baker-Campbell-Hausdorff formula shows exp(X)exp(Y) = exp(X + Y + (1/2)[X,Y] + ...), so the leading correction to naive addition is exactly half the bracket. A caveat: [.,.] is generally NOT associative — that is why a Lie algebra is its own kind of algebra, distinct from an associative ring.