TITLE: The invariant 3-form on a compact Lie group
QUESTION [19 upvotes]: Let $G$ be a compact Lie group. We have the well-known Maurer-Cartan left-invariant and right-invariant $1$-forms $\theta$ and $\bar\theta$ in $\Omega^1(G, \mathfrak{g})$, probably discussed in every Lie theory lectures.
However the canonical bi-invariant closed $3$-form $\chi = \frac{1}{12} (\theta, [\theta, \theta]) = \frac{1}{12} (\bar\theta, [\bar\theta, \bar\theta])$ in $\Omega^3(G)$ may be a little bit less-known. And when I heard of it I had some questions in mind...
1) Are there canonical invariant $5$-forms and higher invariant forms on a $G$?
2) How are they related to Lie algebra cohomology and equivariant cohomology?
3) The construction looks a little bit like $tr(A \wedge dA)$, if we regard $\theta$ as a connection $A$ on the frame bundle of $G$, and use the Maurer-Cartan equation $d\theta = -\frac{1}{2}[\theta, \theta]$. 
Now there is another famous $3$-form: the Chern-Simons form $tr(A \wedge dA + \frac{2}{3} A \wedge A \wedge A)$, and I wonder if these two are somehow related. Does the $tr(A \wedge A \wedge A)$ part vanish here, due to Jacobi identity? 
(and, note there are Chern-Simons 5-forms etc.)
Thank you very much.

REPLY [8 votes]: The 3-form in question is the extension to de Rham cohomology of the group of the canonical semisimple Lie algebra 3-cocycle $\langle - [-,-] \rangle$. 
A Lie algebra cocycle is the same as a closed element in the Chevalley-Eilenberg algebra. If any Lie algebra cocycle is in transgression with an invariant polynomial on the Lie algebra, then this is witnessed by the existence of a corresponding Chern-Simons element in the Weil algebra. The detailed construction is recalled here.
From this simple cohomological construction result most of the phenomena in higher Chern-Simons theory relating these algebraic entities. I have once written up some notes on this with Jim Stasheff and Hisham Sati (from section 6.3 on it discusses what I just mentioned).