Appendix B. Lie’s Third Theorem

Abstract

A finite-dimensional real Lie algebra is the semidirect product of a semisimple subalgebra and the solvable radical, according to the Levi decomposition. As a consequence of this theorem and the correspondence between semidirect products of Lie algebras and semidirect products of simply connected analytic groups, every finite-dimensional real Lie algebra is the Lie algebra of an analytic group. This is Lie's Third Theorem.

Ado's Theorem says that every finite-dimensional real Lie algebra admits a one-one finite-dimensional representation on a complex vector space. This result sharpens Lie's Third Theorem, saying that every real Lie algebra is the Lie algebra of an analytic group of matrices.

The Campbell-Baker-Hausdorff Formula expresses the multiplication rule near the identity in an analytic group in terms of the linear operations and bracket multiplication within the Lie algebra. Thus it tells constructively how to pass from a finite-dimensional real Lie algebra to the multiplication rule for the corresponding analytic group in a neighborhood of the identity.