I am now reading Thom's famous paper Quelques propriétés globales des variétés différentiables. In page 48, Thom used an auxiliary space $K$, which is a principal fiber bundle with base space $K(\mathbb{Z},k)$ and fiber $K(\mathbb{Z},k+4)$, and constructed a map $f\!:MSO(k)\rightarrow K$ extended from the map induced by Thom class $U\in H^k(MSO(k);\mathbb{Z})$. Furthermore, he constructed a map $F=f\times g\!:MSO(k)\rightarrow K\times K(\mathbb{Z}/2,5)$ where $g$ is given by the class $UW_2W_3\in H^{k+5}(MSO(k);\mathbb{Z}/2)$(the image of the product of Stiefel-Whitney classes $W_2W_3\in H^*(BSO(k);\mathbb{Z}/2)$ via Thom isomorphism).
In previous section, Thom shown that $$H^*(K;\mathbb{Z}/2)\cong H^*(\mathbb{Z},k;\mathbb{Z}/2)\otimes H^*(\mathbb{Z},k+4;\mathbb{Z}/2) $$ and he denoted $\iota$ the image of the fundamental class of $H^*(\mathbb{Z},k;\mathbb{Z}/2)$ in $H^{k+4}(K;\mathbb{Z}/2)$ and $\nu$ the image of the fundamental class of $H^*(\mathbb{Z},k+4;\mathbb{Z}/2)\in H^*(MSO(k);\mathbb{Z}/2)$. In mod 2 cohomology calculation of $F$, Thom just wrote that $F^*(\nu)=UW_2^2$ without any explanation. I cannot figure it out why we have such result, can any body tell me how to calculate such $F$? Furthermore, for odd prime $p$, why we have $F^*(\nu)=UP_4\in H^{k+4}(MSO(k);\mathbb{Z}/p)$, where $P_4$ is a Pontrjagin class?
The English translated versioin of this paper can be found in Topological Library. Part 1: Cobordisms and Their Appilications edited by Novikov and Taimanov, translated by Manturov. (There are some typo in this translated version, but they are not big deals.)