Relation graphs of real Cayley–Dickson algebras

The paper presents anticommutativity conditions for elements of arbitrary real Cayley–Dickson algebras, based on which the anticommutativity graphs on equivalence classes of such algebras are classified. Under some additional conditions on the algebras considered, an expression for the centralizer of an element in terms of its orthogonalizer is obtained. Conditions sufficient for this interrelation to hold are provided. Also examples of real Cayley–Dickson algebras in which the centralizer and orthogonalizer of an element are not interrelated in this way are considered.