Geometric interpretation of the map $SO(4) \to SO(3)$

The double cover $\pi:S^3\times S^3\rightarrow SO(4)$ actually provides two homotopoically distinct maps from $S^3 = Sp(1)$ to $SO(4)$, given by restricting $\pi$ to either factor. (One can easily see that these two maps induce different maps on $\pi_3$, so are not homotopic).

The following theorem is contained in

K.Grove-W.Ziller, Lifting group actions and nonnegative curvature, Trans. Amer. Math. Soc. 363 (2011) 2865-2890.

Ziller has a freely accessible version here - see the middle of page $8$.

Theorem If $E\rightarrow M$ is a rank $4$ vector bundle over $M$ (where $M$ is a compact simply connected manifold.), then $\Lambda^2(E) = \Lambda^2(E)_+\oplus\Lambda^2(E)_-$ decomposes into self-dual and antiself dual forms. Then $\Lambda^2(E)_{\pm}$ are the two rank $3$ vector bundles over $M$ corresponding to the two sections above.