= Differential of a Lie group homomorphism preserves Lie brackets
{title2=$df_e([X,Y])=[df_eX,df_eY]$}
= Naturality of the Lie bracket
{synonym}
For a smooth homomorphism of <Lie groups>, naturality gives $f(\exp X)=\exp(df_eX)$. Therefore $\log(f(g))=df_e(\log g)$ 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>.
Back to article page
