Splitting principle for real vector bundles

After pulling a real vector bundle back along a suitable iterated projective or flag bundle, it splits into real line bundles, and the pullback on mod-two cohomology is injective. Identities among Stiefel–Whitney classes can therefore be checked after splitting.

For real vector bundles $E$ and $F$,After applying the splitting principle for real vector bundles, both sides are the product of $1+w_1(L)$ over all line summands.