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.
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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.