Let $k$ be a field, consider the category of $k$-varieties. If a relation among Chern classes of vector bundles holds when all the vector bundles involved split into direct sums of line bundles, does the relation holds in general?
More precisely, let $f=f(X,E_1,\dots,E_l)$ be an expression of Chern classes of vector bundles. Assume the expression behaves well with respect to flat pull backs,i.e. $$p^*(f(X,E_1,\dots,E_l)\frown \alpha)=f(X',p^*E_1,\dots,p^*E_l)$$ when $p:X' \to X$ is flat.
It is well-known that (for example, 3.2.3 of Fulton's Intersection theory) if we can show $f=0$ in the case when each $E_i$ can be written as a successive extensions of line bundles, then $f=0$ is true in general.
Now my question is, if we can merely show that $f=0$ when each $E_i$ is a direct sum of line bundles, do we have $f=0$ in general? Is there a counter-example?
For example, I may want to prove that for a vector bundle $E$ of rank 2, we have
$$c_3(\mathrm{Sym}^2(E))=c_1(E)c_2(E)$$
If the answer to my question is true, then I can prove this formula just in the case $E=L_1\oplus L_2$.
Any help is appreciated, thank you in advance.