Simply connected Lie groups are determined by their Lie algebras

Integration theorem. Let $G$ be a connected simply connected Lie group, let $H$ be a Lie group, and let

be a Lie algebra homomorphism. Then there is a unique Lie group homomorphism $\Phi:G\to H$ such that $d\Phi_e=\varphi$.

Consequently, if $G$ and $H$ are connected and simply connected and their Lie algebras are isomorphic, then $G$ and $H$ are isomorphic as Lie groups.

Existence of a simply connected Lie group integrating a finite-dimensional real Lie algebra is guaranteed by Lie’s third theorem.