A mathematical analysis of zero divisors in real Cayley-Dickson algebras has constructed a loop of six directed links in a zero-divisor graph. The construction applies under the paper's stated setup, including a named condition marked (*), so it is conditional rather than a claim about every such algebra.
The work follows paired algebra elements that can take part in a product equal to the zero element, then studies the resulting zero-divisor and orthogonality graphs. Its focus is on arbitrary real Cayley-Dickson algebras whose components meet additional norm and alternativity conditions.
For a designated family called the main-sequence algebras, the directed zero-divisor graph is obtained from the orthogonality graph by replacing each undirected edge with two directed edges, one pointing each way. This gives the paper a single graph relationship in which the zero-divisor and orthogonality descriptions track the same connections.
From arrows to a double hexagon
Under the stronger conditions used for the orthogonality result, including strong alternativity, zero-product and purity assumptions, together with condition (*), that cycle is an undirected orthogonality-graph hexagon with no chords. In ordinary graph language, no chords means there are no shortcut edges joining non-neighboring points of the six-point loop.
The main-sequence case adds a second construction. The paper describes its double hexagon as six bipartite graphs called K2,2 glued together; each graph has two vertices in each part.
A graph with an algebra underneath
The graph pattern is paired with a multiplication table. The reported table has a block structure: some blocks are antidiagonal, while others resemble the unit-quaternion table. Yet the linear span generated by f0, f1, f2 and f3 is stated not to be associative.
Conditional subalgebra results
The analysis also examines subalgebras generated under tightly specified conditions. One lemma constructs a surjective homomorphism, a structure-preserving map that reaches every element of its target, from a stage-2 Cayley-Dickson algebra with parameters tied to the norms of a and b onto a subalgebra associated with those elements. Under the lemma's conditions, that subalgebra is associative.
A related stage-3 construction maps another parameterized Cayley-Dickson algebra onto a subalgebra associated with a and b, and the lemma says that subalgebra is alternative, an algebraic property named in the result. The supplied analysis also notes that the homomorphism used in these constructions may have a nontrivial kernel, so the results remain tied to the hypotheses under which the maps are built.
Rules for basis-form pairs
Beyond the graph shapes, a theorem gives a conditional classification for basis-form zero divisors. For stage index n at least 1, and for pairs in the specified basis-form domain, it says a pair (a,b) is a zero divisor except in three cases: n is at most 2 and chi is -1; b is a or its negative and chi and gamma sub n are both -1; or b is the unit basis element e0 or its negative and chi equals gamma sub n times the norm of a, with that value equal to -1.
In a separate pure-basis case, the pair made from c and the unit basis element e0 is a zero divisor exactly when chi is 1. When that condition holds, it is orthogonal to the pair made from a and b exactly when the product of a and b equals c multiplied by the norm of b.
The graph also carries a component-level invariant. Among basis-form zero divisors in the same connected component, meaning vertices linked within one part of the graph, the products of the corresponding components are equal up to sign.
For pure components a, b, c and d, the paper gives an orthogonality transform: checking the pair made from a and b against the pair made from c and d is equivalent to checking it against the pair made from d and gamma sub n times c. Gamma sub n is the Cayley-Dickson parameter at the relevant stage.
The conditions set the boundaries
The boundaries of the constructions are explicit. The hexagon results depend on alternativity, zero-product, purity, norm or condition (*) assumptions, and the analysis notes that condition (*) is not true in general. The classification and orthogonality statements are likewise restricted to specified basis-form or purity cases.
This is a symbolic and deductive study, not an empirical sample. Its objects are zero divisors, graph vertices and subalgebras, and its subalgebra results use homomorphism constructions.
The front matter presents the work as arXiv:2608.28176v1 in math.RA, dated 28 Aug 2026. The work was supported by the Russian Science Foundation through project No. 17-11-01124.
Paper data and sources
Original title: Orthogonality graphs of real Cayley-Dickson algebras. Part I: Doubly alternative zero divisors and their hexagons
Authors: Svetlana Zhilina
Journal/Repository: S. Zhilina, Orthogonality graphs of real Cayley-Dickson algebras. Part I: Doubly alternative zero divisors and their hexagons, Int. J. Algebra Comput. 31(4) (2021) 663-689
Status: Peer-reviewed
First online: 2026-08-28
DOI: 10.1142/s0218196721500326
Original paper · Full text