The real $K$-theory of the sphere with an arbitrary involution

We complete the investigation begun in a previous paper to find unitary representations of the non-trivial real $K$-theory elements for the sphere $S^d$ with an involution. Here we consider all involutions except the antipodal involutions. We write down explicit unitaries representing the generators in all cases for $d\le 3$, and for $d>0$ we describe a recipe for generating such unitaries.