Alternativity and reciprocity in the Cayley-Dickson algebra

We calculate the eigenvalue ρ of the multiplication mapping R on the Cayley-Dickson algebra An. If the element in An is composed of a pair of alternative elements in An−1, half the eigenvectors of R in An are still eigenvectors in the subspace which is isomorphic to An−1. The invariant under the reciprocal transformation An × An ∋ (x, y) 7→ (−y, x) plays a fundamental role in simplifying the functional form of ρ. If some physical field can be identified with the eigenspace of R, with an injective map from the field to a scalar quantity (such as a mass) m, then there is a one-to-one map π : m 7→ ρ. As an example, the electro-weak gauge field can be regarded as the eigenspace of R, where π implies that the W-boson mass is less than the Z-boson mass, as in the standard model.