Baker-Campbell-Hausdorff formula
When numbers commute, e^a e^b = e^{a+b}: the exponents just add. Matrices usually do NOT commute, so the product exp(A)exp(B) is some third group element exp(C), and C is generally NOT A + B. The Baker-Campbell-Hausdorff formula tells you exactly what C is — and astonishingly, C is built entirely out of A, B, and their iterated brackets.
The formula reads log(exp A exp B) = A + B + (1/2)[A,B] + (1/12)[A,[A,B]] - (1/12)[B,[A,B]] + ..., an infinite series whose every term is a nested commutator of A and B. So combining two flows produces the naive sum A + B, plus correction terms measuring non-commutativity, the leading one being half the bracket.
This is the precise reason the Lie BRACKET, not the matrix product, is the right operation on a Lie algebra: BCH shows that group multiplication near the identity is completely determined by the bracket structure. Everything about the local group is encoded in g and its [.,.] — that is the formal version of 'a Lie group is captured by its Lie algebra'.
Two caveats. First, if [A, B] = 0 every correction vanishes and you recover C = A + B, the commuting case. Second, the series converges only for A, B small enough (near 0 in g); globally you cannot expect a single closed form. In practice the first couple of bracket terms already give excellent approximations for small steps — the basis of splitting methods in numerical integration.
The leading BCH correction is half the commutator; it disappears exactly when A and B commute, restoring the scalar exponential law.
A clean consequence is the Lie product formula: exp(A+B) = lim_{n->inf} (exp(A/n) exp(B/n))^n. Splitting each flow into n tiny steps and interleaving them makes the bracket corrections vanish in the limit — the rigorous way to 'add' non-commuting generators.