Q:

Stuck in a step of the deduction of cocycle identity

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 $K$).

A:

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.

