Let $V$ be the vector bundle over $BSO(3)$ associated to the adjoint representation of $SO(3).$ Then $V$ does not have a nonzero section. One way to see this is that the Steifel-Whitney class $w_3(V)$ is nonzero.

Question: What about $V \oplus V$ or $V \oplus V \oplus \dots \oplus V?$

- Do these bundles have a nonzero section?
- If not, is there a characteristic class of some sort which obstructs the existence of such a section?

Comments:

The representation given by the direct sum of $n$ copies of the adjoint does not have a one dimensional invariant subspace. Is that the same as (1) above??

The complexification $V \otimes {\mathbb C}$ has trivial third Chern class.

Thanks. Jonathan