I don't understand a passage in the famous article of milnor and kervaire: let $\xi : E \to S^n$ be a vector bundle (of rank $k$) and let $[\xi] \in \pi_{n-1}(SO_k)$ the map associated to $\xi$. Let $X(\xi)$ be the euler class of $\xi$. Prove that the map: $$ [\xi] \mapsto X(\xi) $$ is a homomorphism. ("this is clearly an additive funtion")
The proof consists on three lines in lemma 7 of the article of milnor and kervaire: "a procedure of killing homotopy groups of differentiable manifolds" but I am really not able to understand it. I think that the problem is to understand exactly which definition of euler class is used!
www.maths.ed.ac.uk/~aar/papers/milnorsurg.pdf