I have found interest in the subject of exotic spheres. Particularly, the 7-sphere. To create such a sphere, you glue two copies $\mathbb{D}^4\times \mathbb{S}^3$ along the boundary identifying $(u,v)$ with $(u,u^hvu^j)$ such that $h+j=\pm1$.
Now my question is, the fact that I am trying to construct such a sphere by gluing two 7-disks along the boundary via a diffeomorphism. However, it seems I cannot find such a diffeomorphism explicitly. By diffeomorphism I mean the map $h:\partial \mathbb{S}^6\to\partial \mathbb{S}^6$.
Any help is appreciated.