I am starting to read relative lie algebra cohomology. We define the coboundary operator $d$ from $Hom_K(\wedge^q\mathcal{g}/\mathcal{k}, V)$ to $Hom_K(\wedge^{q+1}\mathcal{g}/\mathcal{k}, V)$ as follows:
$d\eta(X_0\wedge\cdots\wedge X_q)=\displaystyle\sum_{i=o}^{q} (-1)^i X_i.\eta(X_0\wedge \cdots \hat{X_i}\cdots \wedge X_q)+\displaystyle\sum_{0\leq i<j \leq q} (-1)^{i+j}\eta([X_i,X_j] \wedge X_0 \wedge \cdots \hat{X_i} \cdots \hat{X_j} \cdots \wedge X_q)$
where $\eta \in Hom_K(\wedge^q\mathcal{g}/\mathcal{k}, V)$ and $\hat{X_i}$ means that $X_i$ does not appear in the wedge product. I have seen this type of coboundary operators in several different contexts. But, it's very tedious to keep track of all the indices while checking say that $d^2=0$ or in this case, that $d\eta$ intertwines the $K$ module structure on $\wedge^q\mathcal{g}/\mathcal{k}$ and $V$. Can anyone give a nice looking verification of this fact, or at least point to some source where it has been done. Every book I have seen just states these and leaves them for us to prove.