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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. :)

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.

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 $\Pi_f(E) = h^* \prod_j g(p_1(P_j))$ then true for suitable real 2-plane bundles $P_j$ in which some vector bundle $E'$ splits as direct sum such that $E=h^*E'$?

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

I appreciate some literature or concise explaination. :)

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. :)