I am trying to understand a proof from Toda’s paper Cohomology of classifying spaces. The step I am stuck on is at page 96. Here is the setup.

We work with cohomology with $mathbb{F}_2$ coefficients. Let $G$ be a compact Lie group, $Gammasubset G$ be a central subgroup which is isomorphic to either $S_1$ or $mathbb{Z}/2$, let $E_1:=H^*(B^2Gamma)otimes H^*(BG)$ with differential

$$d:=(muotimes 1)circ (1otimes thetaotimes 1)circ (1circ phi),$$

where $phi:H^*(BG)to H^*(BGamma)otimes H^*(BG)$ is the coaction map (i.e. the map induced by the multiplication $Gammatimes Gto G$, which is a group homomorphism as $Gamma$ is central), $mu$ is the multiplication map, and $theta$ is the $mathbb{F}_2$-linear map defined as follows: if

$$H^*(BGamma)=mathbb{F}_2(x_2),qquad H^*(BBGamma)=mathbb{F}_2(z_2,z_3,z_5,dots,z_{2^k+1},dots)$$ (where $z_2=0$ when $Gamma=S^1$), then

$$theta(x_2^{2^i})=z_{2^{i+1}+1}, qquad theta(x_2^j)=0 text{ if $j$ is not a power of $2$}.$$

Simply put, I don’t understand the reasoning at page 96, which from (4.2) deduces Theorem 4.1. Let me be more specific. Assume that $G=SU(n)$, $Gamma=S_1$, and let $q$ be the highest power of two dividing $n$.

At page 96, right before the statement of Theorem 4.1, it is claimed that

$$(*)qquad H^*(E_1,d)=H^*(Cotimes B,d)$$

where $C=mathbb{F}_2(z_3,dots,z_q)$ and $B={ain H^*(BG): d_i(a)=0 text{ for all $igeq q$}}$, where the $d_i$ are the components of $phi$: $phi(a)=sum x^iotimes d_i(a)$. The proof of $(*)$ is supposedly in the few sentences leading up to the statement of Theorem 4.1, but I can’t quite follow it.