This is my first question.

Take a simple, connected, compact, simply connected Lie group $ G$ (dim $G\geq 3$).

The cohomology of $G$ with integer coefficients is
$H^{1,2}(G,\mathbb{Z})\cong 0$, $H^{3}(G,\mathbb{Z})\cong\mathbb{Z}$.

I'm looking for a generator of $H^{3}(G,\mathbb{Z})$.

It should be something of the form $\alpha K(\theta,[\theta,\theta])$, where $\theta$ denotes the left Maurer-Cartan-Form, $K$ is the Killing-Form and $\alpha$ is a scaling factor.