Rank-two complex vector bundles over the four-sphere (${\mathrm{Vect}}_{\mathbb{C}}^2(S^4)\cong \mathbb{Z}$)

The contractible universal frame bundle gives ${\pi }_4(B{U}_2)\cong {\pi }_3(U_2)$ and simple connectedness of $B{U}_2$. The determinant fibration $S{U}_2\to U_2\to S^1$ identifies ${\pi }_3(U_2)$ with ${\pi }_3(S{U}_2)$. Since $S{U}_2\cong S^3$, this is $\mathbb{Z}$. Hence the classification of complex vector bundles by a Grassmannian gives the displayed bijection of isomorphism classes; fixed-rank classes do not acquire a group operation from Whitney sum.