Stuck in a step of the deduction of cocycle identity

Question

In the proof of the cocycle identity, how does the red arrow valid? I'd thought about it for a long while, but can't see the reason.

PS: The author denote the operation in $G$ as $+$ for convenience (even outside the abelian group $K$).

Answer 1

This book appears to be using additive notation for a non-abelian group, where multiplicative notation might be less confusing. The point is that the action of $Q$ on $K$ is given by conjugation in $G$ (this is what the author means when the author says that the extension is "realizing the operators").

It follows that $xf(y,z)$, being $x$ acting on something in $K$, is $$ -\ell(x) + f(y, z) + \ell(x). $$ From here you can get line 2 out of line 1.