the splitting principle
Characteristic classes of a rank-n bundle would be easy to handle if every bundle were just a sum of line bundles, because line bundles are the simple atoms whose classes you understand completely. Most bundles are not such sums, but the splitting principle is a clever trick that lets you pretend they are. It says: for any bundle E over X, there is an auxiliary space F(E) and a map p: F(E) -> X such that pulling E back to F(E) does split into a direct sum of line bundles, and — crucially — the induced map on cohomology p^*: H^*(X) -> H^*(F(E)) is injective.
Concretely F(E) is the flag bundle of E: its points over x are full flags (nested chains of subspaces) of the fiber E_x, and the pullback p^*E acquires a filtration whose quotients are line bundles L_1, ..., L_n, so p^*E is iso to L_1 + ... + L_n. Because p^* is injective, any identity you prove about characteristic classes upstairs, where the bundle is split, descends to a true identity downstairs on X. So the working method is: compute as if E = L_1 + ... + L_n with formal roots x_i = c_1(L_i), the Chern roots; write the total Chern class as a product (1 + x_1)...(1 + x_n); read off c_k as the elementary symmetric polynomial e_k(x_1,...,x_n); and trust that any symmetric expression in the x_i is a genuine class on X.
This is the everyday engine for proving formulas — the Whitney sum formula c(E + E') = c(E)c(E'), the multiplicativity of the Chern character, the definition of the Chern character itself as sum of exp(x_i). The honest caveat: the splitting principle does NOT claim E itself splits — that is usually false — only that it splits after pulling back to a larger space, and that this pullback is faithful enough on cohomology to transport identities back. It is a computational reduction, not a structural decomposition of E.
To prove the Whitney sum formula for c_1 and c_2 of E + E', split each: write c(E) = (1+x_1)(1+x_2) and c(E') = (1+y_1). Then c(E+E') = (1+x_1)(1+x_2)(1+y_1), and multiplying out, the total class is the product c(E)c(E'). Reading degree by degree recovers c_1(E+E') = c_1(E)+c_1(E') and c_2(E+E') = c_2(E)+c_1(E)c_1(E')+c_2(E).
Pretend the bundle is a sum of lines, multiply, read off the symmetric polynomials — the splitting principle guarantees the answer is correct.
The splitting principle does not say E splits; it usually does not. It only says E splits after pullback to the flag bundle, where the pullback map on cohomology is injective, which is enough to verify identities. Confusing 'splits after pullback' with 'splits' is the standard error.