Let $G$ be a connected and simply connected finite-dimensional real Lie group with identity element $e_G$ and [Lie algebra](/page/Lie%20Algebra) $\mathfrak g=T_{e_G}G$. Let $H$ be a finite-dimensional real Lie group with identity element $e_H$ and Lie algebra $\mathfrak h=T_{e_H}H$. If $A:\mathfrak g\to\mathfrak h$ is a homomorphism of real Lie algebras, then there exists a unique smooth Lie [group homomorphism](/page/Group%20Homomorphism) $\varphi:G\to H$ such that

\begin{align*} d\varphi_{e_G}=A. \end{align*}