Duràn wrote down an explicit formula for such map in "Pointed Wiedersehen Metrics on Exotic Spheres and Diffeomorphisms of $S^6$". Geometric properties of that particular map were later studied by various people. For example, it's written down explicitly on page 1 in "Bootstrapping $Ad$-Equivariant Maps, Diffeomorphisms and Involutions" by Duràn and Rigas and Sperança (this link is freely accessible unlike the first one).
Vitali Kapovitch
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