Let $G$ and $H$ be finite-dimensional real Lie groups with identity elements $e_G$ and $e_H$, and let their Lie algebras be $\mathfrak g=T_{e_G}G$ and $\mathfrak h=T_{e_H}H$. If $\Phi:\mathfrak g\to\mathfrak h$ is an isomorphism of real Lie algebras, then there exist open neighbourhoods $U\subset G$ of $e_G$ and $V\subset H$ of $e_H$ and a diffeomorphism $F:U\to V$ such that, for all $g,h\in U$ with $gh\in U$,

\begin{align*} F(gh)=F(g)F(h). \end{align*}