Let $G$ be a compact Lie group and $\mathfrak g$ its Lie algebra.
For any $f$ in the dual space $\mathfrak g^\*$, we can define a skew-symmetric bi-linear form on $\mathfrak g$ by $(A,B)\mapsto f([A,B])$.
Let us say that a differential 2-form on $G$ is a *commutator 2-form* if it has this structure at every point.
We can identify the tangent space by left translations, so commutator 2-forms correspond to functions in $C^\infty(G,\mathfrak g^\*)$.
(The function corresponding to a given 2-form is not generally unique although every function corresponds to some form. For a silly example, consider any abelian group: all commutator 2-forms vanish identically.)


I have some questions about this concept:


* Have commutator 2-forms been studied, perhaps under another name? If yes, where can I read more? I have not managed to find anything in this direction, but I may be missing a crucial keyword in my searches.
* What kinds of commutator 2-forms are closed? In particular, can a non-constant commutator 2-form be closed? If exactness is better understood than closedness, answers in that case are welcome as well.
* If the answer to the previous question is complicated in general, are there examples of non-abelian Lie groups where every non-constant commutator 2-form is closed?



Commutator 2-forms on Lie groups
Let $X$, $Y$, be left invariant vector fields, and let $f$ be a left invariant one-form. By Cartan's formula,
$$(df)(X,Y)=X(\underbrace{f(Y)}\_{\text{const}})-Y(\underbrace{f(X)}\_{\text{const}})-f([X,Y])=-f([X,Y])\;,$$
so your constant commutator $2$-forms are exact.
In the nonconstant case, Cartan's formula together with the Jacobi identity give a simple expression for $d(f([-,-]))$.


