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Splitting principle for real vector bundles

I'm reading the Book of John Roe, Elliptic Operators, Topology and Asymptotic Methods and got stuck 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 $\Pi_f(h^*E) = \prod_j g(p_1(P_j))$ then true for suitable real 2-plane bundles $P_j$ in which $h^*$ splits as direct sum?

iii) What happens in odd dimensions? Has it something to do with choosing $g(0)=1$?

I appreciate some literature or concise explaination. :)

EDIT: I've found a good reference [1]. Namely Proposition 11.2 together with Observation 11.8 gives a satisfying answer for me.

[1] H. Blaine Lawson and Marie-Louise Michelsohn. Spin Geometry.