Let $G$ be a semi simple Lie group (or real reductive), $\mathfrak{g}$ its lie algebra and $B$ its killing form. We can defined the 3-form $k$ by $$k(X,Y,Z)=B([X,Y],Z).$$ with $X,Y,Z\in \mathfrak{g}$. In fact, $k$ is nothing but the structure constants.

It is easy to prove that $k$ is a closed forme on $G$. For example, let $\nabla$ be the connection on $TG$ defined by $\nabla_XY=0$. Its torsion is $T(X,Y)=-[X,Y]$. Then $$d=e^i\wedge\nabla_{e_i}+i_{T}.$$ where $e_i$ is a base of $\mathfrak{g}$ and $e^i$ is the dual base. It is easy to verifier $e^i\wedge\nabla_{e_i}k=0$ and $i_{T}k=0$. So $dk=0$.

Can someone give some explanations of the 3 closed form $k$? For example,

- What is the topology mean of its cohomology class $[k]\in H^3(G)$.
- How about the case $[k]=0$ or $[k]\neq0$?
- If $G$ is a compact semi simple Lie group, can we say more about the form $k$?