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 ).