Simply connected Lie groups are determined by their Lie algebras

A central “uniqueness” principle in Lie theory is:

Theorem (uniqueness for simply connected groups). Let $G$ and $H$ be connected simply connected Lie groups. If their Lie algebras are isomorphic,

(as Lie algebras; see Lie algebra isomorphism), then $G$ and $H$ are isomorphic as Lie groups.

More precisely, if $\varphi:\mathfrak g\to\mathfrak h$ is a Lie algebra homomorphism between $\mathfrak g=\mathrm{Lie}(G)$ and $\mathfrak h=\mathrm{Lie}(H)$, then there exists a unique Lie group homomorphism $\Phi:G\to H$ such that $d\Phi_e=\varphi$ (see differential is a Lie algebra homomorphism). When $\varphi$ is an isomorphism, $\Phi$ is an isomorphism.

Existence of a simply connected Lie group integrating a given finite-dimensional Lie algebra is guaranteed by Lie’s third theorem. The uniqueness above explains why, in practice, one often works “purely algebraically” at the level of Lie algebras and then passes to a canonical global group by taking the simply connected (universal cover) Lie group (see existence of universal covering groups).