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How about thinking about framed cobordism, which is this case gives an isomorphism between $\pi_4(S^3)$ and the group of cobordism classes of normally framed 1-manifolds in $S^4$. Since your map is constructed geometrically, it probably has a regular value. Pull this back to a collection of disjoint circles in $S^4$, forming a trivial link, with framings of their normal bundles. Since $\pi_1SO(3)$ has order 2, just count up the number of circles with nontrivial framing to see whether this number is odd or even.