The Baker–Campbell–Hausdorff formula

Choose exponential coordinates near the identity, where a local logarithm exists. The logarithm of exp X times exp Y begins with X+Y and then one half [X,Y]. Reversing X and Y reverses this second-degree term. Commuting generators have no bracket correction and recover the ordinary additive law. For noncommuting generators, the discrepancy is second order in a simultaneous small scaling of X and Y, which explains why a first-order linear picture can hide a genuine order effect.

The next terms involve brackets of length three. Their coefficients and signs can be checked by expanding matrix exponentials and the local logarithm to the same degree. A finite truncation is an approximation whose error depends on the scale and chosen norm. Reporting only the truncated expression as an exact product would discard precisely the higher nested commutators that distinguish a general Lie algebra from a simpler one.

In the formal setting, X and Y are symbols and equality means that coefficients agree at every degree. Convergence is not part of that definition. In a finite-dimensional real Lie algebra, a norm and analytic local group structure give a genuine neighborhood where the series converges and agrees with the local logarithm. The size of an admissible neighborhood depends on the algebra and norm. A branch of logarithm must remain chosen consistently in a matrix calculation.

For a two-step nilpotent Lie algebra, every bracket of length three vanishes, leaving only X+Y plus one half [X,Y]. On its connected simply connected Lie group, exp is a global diffeomorphism and this finite polynomial describes the group law globally in exponential coordinates. Without the simply connected qualification, an additional discrete central quotient can identify different exponential coordinates. Termination of the polynomial and injectivity of the coordinate map are separate facts.

Strictly upper-triangular three-by-three matrices give a concrete two-step nilpotent example. The first upper diagonal positions do not commute, but their commutator lands in the top-right position, which commutes with the whole algebra. The group of upper-triangular matrices with diagonal ones is diffeomorphic to real three-space. Its matrix exponential and logarithm are finite polynomials, so the BCH calculation can be checked exactly even with parameters larger than a generic local convergence neighborhood.

This model is useful for testing order-sensitive code and for understanding the central displacement generated by two horizontal motions. It does not justify truncating BCH after one commutator in a rotation algebra or any other non-nilpotent example. A reliable calculation states whether it is exact because brackets terminate, approximate because a small parameter controls a remainder, or merely formal because no convergence question is being asked. Those categories lead to different numerical and geometric claims.