Lie's third theorem

In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra ${\displaystyle {\mathfrak {g}}}$ over the real numbers is associated to a Lie group ${\displaystyle G}$. The theorem is part of the Lie group–Lie algebra correspondence.

Historically, the third theorem referred to a different but related result. The two preceding theorems of Sophus Lie, restated in modern language, relate to the infinitesimal transformations of a group action on a smooth manifold. The third theorem on the list stated the Jacobi identity for the infinitesimal transformations of a local Lie group. Conversely, in the presence of a Lie algebra of vector fields, integration gives a local Lie group action. The result now known as the third theorem provides an intrinsic and global converse to the original theorem.