TITLE: Intuition/idea behind a proof of the splitting principle?
QUESTION [13 upvotes]: The splitting principle is as follows.

Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map $p^*: K^*(X)  \to K^*(F(E))$ is injective and $p^*(E)$ splits as the sum of line bundles.

My question is, what is the idea/intuition behind the proof of the splitting principle?

REPLY [17 votes]: Perhaps my very short (4 pages plus bibliography) paper ``A note on the splitting principle'' http://www.math.uchicago.edu/~may/PAPERS/Split.pdf
may be illuminating. It shows that the splitting principle can be viewed as a statement about the reduction of the structural group of a $G$-bundle $\xi$ from $G$ to a maximal torus $T$, where $G$ is a compact Lie group. It applies more generally than in just the usual examples. One starts with the bundle $BT\to BG$ with fiber $G/T$. For a $G$-bundle over $X$ classified by $f\colon X\to BG$, one has a pullback bundle $q\colon Y\to X$ with fiber $G/T$ together with a reduction of the structure group of $q^*\xi$ to $T$. When $H^*(BG;R)$ is concentrated in even degrees, $q^*\colon H^*(X;R)\to H^*(Y;R)$ is a monomorphism. That is easily seen to imply the splitting theorem as usually stated, and many variants thereof.
As stated and explained briefly in the paper, the argument adapts to $K$-theory.