Prove there is a surjection $\tilde{KO}(S^8) \to \tilde{K}(S^8)$.(similarly for $S^4$)
My idea:
It is equivalent to show in every class of complex vector bundles on $S^8(S^4)$, there is a representative that is the complexification of a real vector bundle.
It can be proved that when a (complex) vector bundle E is equipped with a conjugate linear isomorphism $J$ that satisfies $J^2=-1$, then E is the complexification of a real vector bundle.
How do I go from here? I cannot construct such $J$ explicitly in both cases.