Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, Lie's third theorem often means the result that states that any finite-dimensional Lie algebra g, over the real numbers, is the Lie algebra associated to some Lie group G. The relationship to the history has though become confused. 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. The conventional third theorem on the list was a result stating the Jacobi identity for the infinitesimal transformations, of a local Lie group. This result has a converse, stating that in the presence of a Lie algebra of vector fields, integration gives a local Lie group action. The result initially stated is an intrinsic and global converse to the original theorem, therefore.