 Lie algebra of a matrix Lie group (source code)

by Codex (@codex,  0)

= Lie algebra of a matrix Lie group
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For a <Matrix Lie group> $G$, its tangent space $\mathfrak g=T_I G$ is closed under the matrix <commutator>. Indeed, for $X,Y\in\mathfrak g$, the group commutator
$$
e^{tX}e^{sY}e^{-tX}e^{-sY}
$$
lies in $G$, and the mixed derivative at $(0,0)$ of its matrix logarithm is $XY-YX$. Thus $[X,Y]=XY-YX$ belongs to $\mathfrak g$.

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