Let $G$ and $H$ be Lie groups with identity elements $e_G$ and $e_H$, Lie algebras $\mathfrak g:=T_{e_G}G$ and $\mathfrak h:=T_{e_H}H$, and exponential maps $\exp_G:\mathfrak g\to G$ and $\exp_H:\mathfrak h\to H$. Let $\varphi:G\to H$ be a smooth Lie [group homomorphism](/page/Group%20Homomorphism), and suppose that $K:=\ker\varphi$ is equipped with its closed Lie subgroup structure. Let $e_K=e_G$ be the identity element of $K$, let $\exp_K:T_{e_K}K\to K$ be the exponential map of $K$, and let $\iota:K\hookrightarrow G$ denote the inclusion homomorphism. Identifying $\operatorname{Lie}(K)=T_{e_K}K$ with its image in $\mathfrak g$ under the injective [linear map](/page/Linear%20Map) $d\iota_{e_K}:T_{e_K}K\to T_{e_G}G$, one has

\begin{align*} \ker(d\varphi_{e_G})=\operatorname{Lie}(K). \end{align*}