Blakers-Massey elements and exotic diffeomorphisms of and via geodesics

We use the geometry of the geodesics of a certain left-invariant metric on the Lie group to find explicit related formulas for two topological objects: the Blakers-Massey element (a generator of ) and an exotic (i.e. not isotopic to the identity) diffeomorphism of (C. E. Durán, 2001). These formulas depend on two quaternions and their conjugates and we produce their extensions to the octonions through formulas for a generator of and exotic diffeomorphisms of , thus giving explicit gluing maps for half of the 15-dimensional exotic spheres expressed as the union of two 15-disks.