= Lie bracket from local group commutators
{c}
{title2=$[X,Y]=\partial_s\partial_t\log(e^{sX}e^{tY}e^{-sX}e^{-tY})|_{s=t=0}$}
In a <local exponential chart>, the mixed second differential of the <group commutator> defines a <bilinear map> on the <tangent space> at the identity. Inversion after swapping the two group elements proves antisymmetry. Differentiating conjugation gives $[X,Y]=\operatorname{ad}(X)Y$; applying <naturality of the Lie bracket> to the <Adjoint representation of a Lie group> then proves the <Jacobi identity>. For a matrix group this recovers $XY-YX$ by expanding through the mixed term.
Back to article page
