I am reading the famous paper by Dold and Whitney "Classification of Oriented Sphere Bundles Over A 4-Complex".

I'll state their theorem for the case of SO(3) bundles

**Classification Theorem**:Let $B_1, B_2$ be principal $SO(3)$ bundles over a complex $K$ of dimension at most 4. let $h_i:K\to G_n\simeq BSO(3)$ be the classifying map for $B_i$.
Assume they have the same second Stiefel-Whitney class $w_2(B_1)=w_2(B_2)=: w_2$ (then we can assume that $h_1$ and $h_2$ agree on the 3-skeleton $K^3$).
Let $d(h_1,h_2) \in H^4(K;\pi_4(G_n))$ be the difference cocycle cohomology class.

The bundles $B_1,B_2$ are equivalent iif
there exists a cohomology class $x\in H^1(K,\mathbb{Z}/2)$ such that

$$(1) \quad \quad d(h_1,h_2) = \beta x \cup \beta x + \beta (x\cup w_2)$$

where $\beta$ is the Bockstein homomorphism associated with the coefficient sequence $0\to \mathbb{Z} \overset{2\cdot}{\to} \mathbb{Z}\to \mathbb{Z}/2\to 0$ and $\pi_3(G_n)$ has been identified with $\mathbb{Z}$.

Usually we require the cohomology class of the cocycle to $d(h_1,h_2)$ to be $0$, why in (1) they require that condition instead?

**Explanation of the question**
I have seen stated in others book on obstruction theory that $h_1,h_2:K^n\to Y$ are homotopic rel to $K^{n-2}$ iif $[d(h_1,h_2)]=0$ (under the hypothesis of $Y$ being simple, I don't know if this is the point) this seems to be very different from $(1)$.

**Cohomology operations**. Dold & Whitney appeal to a theorem from Eilenberg MacLane's paper "On the groups $H(\pi,n),\quad III:$ Operations and Obstructions", theorem 14.2.
I looked into it and indeed they don't require the difference cocycle to be null-cohomologous.
I only explained myself that the RHS of $(1)$ comes from the cohomology operation induced by the $k$-invariant (I guess Postnikov invariant) of the target space.

Can someone explain better the setup and theorem 14.2 in Eilenberg MacLane's paper?