Lie's third theorem

In mathematics, Lie's third theorem states that any finite-dimensional Lie algebra g, over the real numbers, is the Lie algebra associated to some Lie group G.

There were (naturally) two other preceding theorems, of Sophus Lie. Those relate to the infinitesimal transformations of a transformation group acting on a smooth manifold. But, in fact, that language is anachronistic. The manifold concept was not clearly defined at the time, the end of the nineteenth century, when Lie was founding the theory.