Lie group homomorphism ($F:G\to H,F(ab)=F(a)F(b)$)

A smooth homomorphism of Lie groups has differential $D_{e}F$ at the identity. Differentiating $F\circ L_g=L_{F(g)}\circ F$ makes the corresponding left-invariant vector fields $F$-related. Their commutator action on pulled-back functions shows that the differential of a Lie group homomorphism preserves Lie brackets. The map need not be injective or surjective.