# reference request: lie algebra-lie group

I am looking for a reference where I can find a (relatively) elementary and self contained proof of the fact that all real, finite dimensional Lie algebras are the Lie algebra of some Lie group.

Thank you

• Have you tried P. J. Olver (amazon.com/…*Version*=1&*entries*=0) – MrYouMath May 30 '16 at 15:19
• The fact you're looking for is known as Ado's theorem – Ben Grossmann May 30 '16 at 15:21

The statement is known as Lie's third Theorem: Every finite-dimensional real Lie algebra $L$ is integrable, that is, there exists a Lie group $G$ with $Lie(G)\cong L$. For a proof see, for example, the note by J. Ebert Van Est's exposition of Cartan's proof of Lie's third theorem, and the references.

Remark: Lie's third theorem is also a corollary to Ado's theorem, see here. However, I think that the proof of Ado's theorem is perhaps more difficult to understand than the proof of Lie's third theorem.