To define the Cohomology (with values in $\mathbb{C}$) on a lie algebra $L$, we define a coboundary map $\delta:\Lambda^n(L)\to \Lambda^{n+1}(L)$. There is a general formula for the coboundary map but I'm only interested in one and two-forms. Say $\eta\in \Lambda^1(L)$ and $\omega\in \Lambda^2(L)$ and $X,Y,Z\in L$ are arbitrary. Then $$ \delta \eta(X,Y) = X\eta(Y)-Y\eta(X) -\eta([X,Y]) $$ $$ \delta\omega(X,Y,Z)=\left\{X\omega(Y,Z)+Y\omega(Z,X)+Z\omega(X,Y)\right\} - \left\{\omega([X,Y],Z)+\omega([Y,Z],X)+\omega([Z,X],Y)\right\} $$ So my understanding is a closed cocyle is one such that all of the right hand side of these equations is zero. But I have repeatedly seen people referring to $$ \omega([X,Y],Z)+\omega([Y,Z],X)+\omega([Z,X],Y)=0 $$ as "2-cocyle condition" and say any 2-form satisfying the 2-cocyle condition is in $Z^2(L)$ (the space of 2-cocyles). Why is the rest $X\omega(Y,Z)+Y\omega(Z,X)+Z\omega(X,Y)$ safe to ignore? Or am Im missing something?
In the same spirit, it is said that $\omega\in B^2(L)$ is a cobounday if $\omega(X,Y)=\eta([X,Y])$ for some 1-cochain. So again why are we throwing $X\eta(Y)-Y\eta(X)$ out?
If the context of this question is important I am reading about theory of Central Extensions of Lie Algebras. I think I'm overlooking some property...
You are considering $\mathbb{C}$ as a trivial $L$ module. So for any $X \in L$ and any 1-cocyle $\eta$ the value of $ X\eta(Y)$ is 0. This is why you ignore all cross terms.