You can find below the proof due to G.M; Tuynman, of the third Lie theorem.
The proof is similar to using the Ado's theoremof Ado, but requires an 'advanced' result:
the fact that for a simply connected Lie group $G$ not only the first de Rham cohomology space $H^1(G)={0}$ but also $H^2(G)={0}$.
In the Godbillon book, the Künneth theorem that identifies the cohomology of product manifolds with the tensor product of their cohomologies: $$K : H(M)\otimes H(N)\rightarrow H(M\times N)$$ If $M$ is compact, $K$ is an isomorphism. There are two things
I can not understand, among others:
1) Godbillon consider $d=K^{-1}\circ\mu^\star$, where $\mu$ is the multiplication on the Lie group $G$, and $K$ is invertible, the compactness is it essential for $K$ to be an isomorphism?!
2) Godbillon theorem shows in his page 202, with $d$, if $G$ is connected then the smallest integer $q>0$ such that $H^q(G)\neq 0$ is odd. First, $G$ must be compact? Why $H^1(G)=0$? This is not posted a consequence of the above theorem where compactness is essential!related question in math.stackexchange.com
http://math.stackexchange.com/questions/56899/elementary-proof-of-the-third-lie-theorem
