(Unstable) cohomological operations are in bijection with cohomology classes of Eilenberg–MacLane spaces. Rings $H^\bullet(K(\mathbb Z;n);\mathbb Z)$ can be computed step by step by considering spectral sequences of Serre fibrations $K(\mathbb Z;n-1)\to\ast\to K(\mathbb Z;n)$.
But how to extract an explicit description of an operation from such computation?
- Consider LSSS of fibration $\mathbb CP^\infty\cong\Omega K(\mathbb Z;3)\to\ast\to K(\mathbb Z;3)$: let $t$ be the generator of $H^2(\mathbb CP^\infty)$ and let $u\in H^3(K(\mathbb Z;3);\mathbb Z)$ be the fundamental class; then $d_3(t^3)=3t^2u$ and $d_5(t^2u)$ gives a 3-torsion element in $H^8(K(\mathbb Z;3);\mathbb Z)$.
- The corresponding cohomological operation is the following «Massey cube»: if $a$ is an integral $3$-cocycle, $[a,a]=0\in H^6$ (where $[-,-]$ is the suppercommutator), so $[a,a]=db$ (where $b$ is some cochain); define $\langle a\rangle^3:=[a,b]\in H^8$.
So concrete question is: suppose $\gamma=d_i(\alpha\beta)$ (in LHSS for $K(\mathbb Z;n-1)\to\ast\to K(\mathbb Z;n)$; $\alpha\in H^\bullet(K(\mathbb Z;n-1))$, $\beta,\gamma\in H^\bullet(K(\mathbb Z;n))$) and we already know opeations corresponding to $\alpha$ and $\beta$; how to construct the operation corresponding to $\gamma$?