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  "body": "\n## Statement\n\nLet $G$ be a real [[Lie group|7JBRGCVJRB]] with Lie algebra $\\mathfrak{g}$ and [[exponential map|H3QX1ZFC8K]] $\\exp \\colon \\mathfrak{g} \\to G$.\nThere is a neighbourhood $U$ of $0$ in $\\mathfrak{g}$ such that for all $X, Y \\in U$,\n$$\\exp(X)\\exp(Y) = \\exp(Z(X, Y)),$$\nwhere $Z(X, Y)$ is the sum of a convergent series\n$$Z(X, Y) = X + Y + \\tfrac12 [X, Y] + \\tfrac1{12}[X, [X, Y]] - \\tfrac1{12}[Y, [X, Y]] - \\tfrac1{24}[Y, [X, [X, Y]]] + \\cdots.$$\nEach term of the series is a rational multiple of an iterated Lie bracket of $X$ and $Y$.\nThe coefficients do not depend on $G$.\n\n## Remarks\n\nThe 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.\nDynkin gave an explicit formula for all the coefficients.\nConvergence for $X$ and $Y$ near $0$ then follows from estimates on the norms of iterated brackets.\nFor matrices, $e^X e^Y = e^{Z(X, Y)}$ with the [[matrix exponential|020M04FN1H]] when $X$ and $Y$ have small enough norm.\nIf $X$ and $Y$ commute, all brackets vanish and the formula reduces to $e^X e^Y = e^{X + Y}$.\nIf $[X, Y]$ commutes with both $X$ and $Y$, as in the Heisenberg group, the series stops after the first bracket: $e^X e^Y = e^{X + Y + \\frac12[X, Y]}$.\nWithout the smallness condition the formula can fail.\nIn $SL_2(\\R)$, take $X = \\begin{pmatrix} 0 \u0026 -\\pi \\\\ \\pi \u0026 0 \\end{pmatrix}$ and $Y = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$; then $e^X = -I$ and $e^X e^Y = \\begin{pmatrix} -1 \u0026 -1 \\\\ 0 \u0026 -1 \\end{pmatrix}$, which is not the exponential of any matrix of trace $0$.\n\n## Explanations\n\n### L1 human\n\nFor numbers, $e^x e^y = e^{x + y}$.\nFor matrices, and in Lie groups generally, this rule fails because the order of multiplication matters.\nThe formula says how to repair it: the exponent of the product is $X + Y$ plus correction terms built only from commutators $[X, Y] = XY - YX$ and commutators of commutators.\n\n### L2 human\n\nThe formula shows that the group law near the identity is determined by the Lie bracket alone.\nThis 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.\nFor 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.\nRelated 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.\n",
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