I'm reading the Book of John Roe, Elliptic Operators, Topology and Asymptotic Methods and got stuck at Lemma at Lemma 2.27.

i) How does this lemma show that a real vector bundle can be given by a pullback of direct sums of plane bundles?

ii) Assuming i) is clear, is this equation $\Pi_f(E) = \prod_j g(p_1(P_j))$ then true for suitable real 2-plane bundles? What happens in odd dimensions?

I appreciate some literature or concise explaination. :)