i am a little bit confused about how to obtain the second line.

i have that $\kappa([A \wedge \delta A] \wedge A) = \kappa([\delta A \wedge A] \wedge A)$ by antisymmetry of $\wedge$ and $[,]$

but i get that $\kappa([A \wedge A] \wedge \delta A) = -\kappa(\delta A \wedge [A \wedge A]) = -\kappa([\delta A \wedge A] \wedge A)$ by cyclicity of $\kappa$ and antisymmetry of $\wedge$ and $[,]$

but this would make the term in question $\kappa(\frac{2}{3} \delta A \wedge A \wedge A)$, which I don't think is correct

in indices i also get this result

okay i think the above expressions are just wrong (the variations i mean)

wait nvm...