Baker–Campbell–Hausdorff formula

Remarks

The formula is first proved for formal power series in two non-commuting variables, where one shows that $\log(e^X e^Y)$ is a series of Lie polynomials. Dynkin gave an explicit formula for all the coefficients. Convergence for $X$ and $Y$ near $0$ then follows from estimates on the norms of iterated brackets. For matrices, $e^X e^Y = e^{Z(X, Y)}$ with the matrix exponential when $X$ and $Y$ have small enough norm. If $X$ and $Y$ commute, all brackets vanish and the formula reduces to $e^X e^Y = e^{X + Y}$. If $[X, Y]$ commutes with both $X$ and $Y$, as in the Heisenberg group, the series stops after the first bracket: $e^X e^Y = e^{X + Y + \frac12[X, Y]}$. Without the smallness condition the formula can fail. In $SL_2(\R)$, take $X = \begin{pmatrix} 0 & -\pi \\ \pi & 0 \end{pmatrix}$ and $Y = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$; then $e^X = -I$ and $e^X e^Y = \begin{pmatrix} -1 & -1 \\ 0 & -1 \end{pmatrix}$, which is not the exponential of any matrix of trace $0$.

Going deeper

For a student

The formula shows that the group law near the identity is determined by the Lie bracket alone. This is the key step in proving that a Lie algebra homomorphism integrates to a local homomorphism of groups, and in building a local Lie group from a Lie algebra. For nilpotent Lie algebras the series is finite, so it defines a group law on the whole Lie algebra; this gives the simply connected nilpotent Lie groups. Related formulas, such as the Lie product formula $e^{X + Y} = \lim_{n \to \infty} (e^{X/n} e^{Y/n})^n$, also express the interplay between the group and the bracket.