Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Fix a basis $e_1,\dots,e_n$ of $\mathfrak{g}$ and let $e^1,\dots,e^n$ be its dual basis. We also use $e^i$ to denote the left-invariant 1-form on $G$ obtained from $e^i$. It seems that I can prove the following identity by brute force: \begin{equation}\sum_{i,j}de^i\wedge de^j\cdot\kappa(e_i,e_j)=0,\end{equation} where $d$ is the exterior derivative and $\kappa$ is the Killing form.
I would like to know how can one tell this identity holds from a glimpse. I believe there must be some easy way to explain such a simple expression but I just cannot figure it out. Thanks!