Lie group-Lie algebra correspondence

Every Lie algebra corresponds to a single simply connected Lie group.

The Baker-Campbell-Hausdorff formula basically defines how to map an algebra to the group.

Bibliography:

Lie Groups, Physics, and Geometry by Robert Gilmore (2008) 7.2 "The covering problem" gives some amazing intuition on the subject as usual.

Example at: Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation".

Example at: Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation".

Furthermore, the non-compact part is always isomorphic to ${\mathbb{R}}^n$, only the non-compact part can have more interesting structure.

The most important example is perhaps $SO(3)$ and $SU(2)$, both of which have the same Lie algebra, but are not isomorphic.

This simply connected is called the universal covering group.

E.g. in the case of $SO(3)$ and $SU(2)$, $SU(2)$ is simply connected, but $SO(3)$ is not.

The unique group referred to at: every Lie algebra has a unique single corresponding simply connected Lie group.