Alternativity and reciprocity in the Cayley-Dickson algebra

We calculate the eigenvalue p of the multiplication mapping R on the Cayley-Dickson algebra A n . If the element in An is composed of a pair of alternative elements in A n-1 , half the eigenvectors of R in An are still eigenvectors in the subspace which is isomorphic to A n-1 . The invariant under the reciprocal transformation An x An 3 (x, y) → (-y, x) plays a fundamental role in simplifying the functional form of p. 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 → p. 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.