Relation Graphs of the Sedenion Algebra

Let S denote the algebra of sedenions and let Γ_O(S) denote its orthogonality graph. One can observe that every pair of zero divisors in S generates a double hexagon in Γ_O(S). The set of vertices of a double hexagon can be extended to a basis of S that has a convenient multiplication table. The set of vertices of an arbitrary connected component of Γ_O(S) is described, and its diameter is found. Then, the bijection between the connected components of Γ_O(S) and the lines in the imaginary part of the octonions is established. Finally, the commutativity graph of the sedenions is considered, and it is shown that all the elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies in between 3 and 4.