Generating varieties, Bott periodicity and instantons

Let G be the classical group and let Mk(G) be the based moduli space of G-instantons on S4 with instanton number k. It is known that Mk(G) yields real and symplectic Bott periodicity, however an explicit geometric description of the homotopy equivalence has not been known. We consider certain orbit spaces in Mk(G) and show that the restriction of the inclusion of Mk(G) into the moduli space of connections, which, in turn, is explicitly described by the commutator map of G. We prove this restriction satisfies a triple loop space version of the generating variety argument of Bott (1958) [5], and it also gives real and symplectic Bott periodicity. This also gives a new proof of real and symplectic Bott periodicity.