Let $E \to X$ be a complex vector bundle of rank $r$. There exists a space $\pi:Y\to X$ such that $\pi^*:H^*(X)\to H^*(Y)$ is injective and \begin{align*} \pi^*E \cong L_1\oplus \cdots \oplus L_r \end{align*} for complex line bundles $L_1,\dots,L_r$ over $Y$. Therefore universal polynomial identities among Chern classes may be checked after replacing $E$ by a direct sum of line bundles.