Logarithmic chart of a matrix Lie group

For a matrix Lie group $G$ with Lie algebra $\mathfrak{g}=T_{I}G$, choose neighborhoods on which $\exp :\mathfrak{g}\to G$ and $\log :G\to \mathfrak{g}$ are inverse. Left translation gives a chart near $g\in G$ byThe Baker--Campbell--Hausdorff formula expresses multiplication smoothly in these coordinates.