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.
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.