I am trying to understand lemma 2.26 from http://www.math.ru.nl/~solleveld/scrip.pdf
I am coserned about calculation of $E^{p, q}_1$. If $\mathfrak{h}$ is Lie ideal than everything is fine. But here $\mathfrak{h}$ is just subalgebra, I am in trouble.
My calculation problem is follows. For notation simplicity let me work with homology (take dual complex). The filtration on homology defines as $$ F^p C^r = \Lambda^p \mathfrak{h} \wedge \Lambda^{(r-p)} \mathfrak{g}$$
We should calculate $d(h_1 \wedge \dots \wedge h_p \wedge g_1 \wedge \dots \wedge g_{r-p}) \in F^p C^r / F^{p+1} C^r$. Lemma would be true if it is equal to $d(h_1 \wedge \dots \wedge h_p )\wedge g_1 \wedge \dots \wedge g_{r-p}$. For this we need
$$(-1)^{i+j} [g_i , g_j] \wedge h_1 \wedge \dots \wedge h_p \wedge g_1 \wedge \dots \wedge \hat{g}_i \wedge \dots \wedge \hat{g}_j \wedge \dots \wedge g_{r-p} \in F^{p+1} C^r$$
$$(-1)^{i+j} [h_i , g_j] \wedge h_1 \wedge \dots \wedge \hat{h}_i \wedge \dots \wedge h_p \wedge g_1 \wedge \dots \wedge \hat{g}_j \wedge \dots \wedge g_{r-p} \in F^{p+1} C^r$$
First is true. Second is true is true only if $\mathfrak{h}$ is ideal.