The relationship between Lie groups and Lie algebras can be summarized as follows. Each Lie group determines a Lie algebra over (concretely, the tangent space at the identity). Conversely, for every finite-dimensional Lie algebra , there is a connected Lie group with Lie algebra . This is Lie's third theorem; see the Baker–Campbell–Hausdorff formula. This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra are locally isomorphic, and more strongly, they have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) have isomorphic Lie algebras, but SU(2) is a simply connected double cover of SO(3).
The relationship between Lie groups and Lie algebras can be summarized as follows. Each Lie group determines a Lie algebra over (concretely, the tangent space at the identity). Conversely for every finite-dimensional Lie algebra , there is a connected Lie group with Lie algebra . This is Lie's third theorem; see the Baker–Campbell–Hausdorff formula. This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra are locally isomorphic, and more strongly, they have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitar,y group SU(2) have isomorphic Lie algebras, but SU(2) is a simply connected double cover of SO(3).