Homotopy sets of multiplications on spheres and their algebraic structure (Q1900003)

In an earlier paper [ibid. 48, 267-294 (1993; Zbl 0813.55005)] the author introduced an algebraic structure on sets of homotopy classes \([X,S^n]\), given by the reflecting product \(\bullet\) on spheres \(S^n\) \((x \bullet y : = - y + 2 \langle x; y \rangle x\) where \(x,y \in S^n)\), which makes these sets into symmetric groupoids. In this article the algebraic structure of the symmetric groupoids \(([S^n \times S^n; S^n], \bullet)\) for \(n \neq 1, 3, 7\) is determined. Here a symmetric groupoid is a set \(R\) with a multiplication \(\bullet : R \times R \to R\) satisfying \(x \bullet x = x\); \(x \bullet (x \bullet y) = y\); \(x \bullet (y \bullet z) = (x \bullet y) \bullet (x \bullet z)\). The paper is very technical. (revised by eds.)