Differential of a Lie group homomorphism preserves Lie brackets ($d{f}_e([X,Y])=[d{f}_{e}X,d{f}_{e}Y]$)

For a smooth homomorphism of Lie groups, naturality gives $f(\exp X)=\exp (d{f}_{e}X)$. Therefore $\log (f(g))=d{f}_e(\log g)$ locally. Apply this to the group commutator and differentiate its two curve parameters to obtain the displayed identity for the Lie bracket from local group commutators. In particular the differential of a group representation is a Lie algebra representation.