@johncarlosbaez i looked at the proof which equates Lie algebra cohomology with de Rham cohomology for (G) a connected compact Lie group to see where the connectedness of (G) is being used exactly. As far as I can tell, this condition is needed so that there exists a curve joining every element (g\in G) to the identity element (e\in G). This implies that can (L_g) (left translation by (g\in G)) is homotopic to (L_e = id). From this one can construct a chain homotopy to show that the inclusion (\iota: \Omega^\bullet_L(G)\hookrightarrow\Omega^\bullet(G)) induces an isomorphism between cohomology of left-invariant forms and de Rham cohomology of all forms. Compactness is needed not only for integration, but also to have a finite partition of unity. The chain homotopy is then a finite sum of linear maps (\Omega^k(G) \rightarrow \Omega^{k-1}(G)) using the finite partition of unity. The proof is not hard hard but also not trivial. I’ve been curious about this result for awhile so it’s interesting to go through the proof and learn new techniques along the way.
Also the same proof can be used to show that if a compact connected Lie group (G) acts on a manifold (M) (from the left), then [H^\bullet_{dR}(M)\simeq H^\bullet_G(M)] where (H^\bullet_G(M)) is the cohomology of (G)-invariant forms on (M).