On $\mathrm{SL}(3,\mathbb{R})$-actions on $4$-spheres

We construct a natural, continuous $\mathrm{SL}(3,\mathbb{R})$-action on $S^{4}$ which is an extension of the $\mathrm{SO}(3)$-action $\psi$ of Uchida. The construction is based on the Kuiper theorem asserting that the quotient space of $\mathbb{C}P(2)$ by complex conjugation is $S^{4}$. We also give a new proof of the Kuiper theorem.