A minimal and non-alternative realisation of the Cayley plane

TL;DR

This paper introduces a simpler, more economical algebraic construction of the Cayley plane using paraoctonions and Okubo algebra, challenging the traditional octonionic approach and highlighting the Okubo algebra's foundational role.

Contribution

The study demonstrates that the Cayley plane can be realized through weaker algebraic structures like paraoctonions and Okubo algebra, expanding understanding of its algebraic foundations.

Findings

The Cayley plane can be defined using paraoctonions and Okubo algebra.

The real Okubo algebra has SU(3) as automorphism group, unlike octonions with G2.

The projective plane over the Okubo algebra is isomorphic to the octonionic projective plane.

Abstract

The compact 16-dimensional Moufang plane, also known as the Cayley plane, has traditionally been defined through the lens of octonionic geometry. In this study, we present a novel approach, demonstrating that the Cayley plane can be defined in an equally clean, straightforward and more economic way using two different division and composition algebras: the paraoctonions and the Okubo algebra. The result is quite surprising since paraoctonions and Okubo algebra possess a weaker algebraic structure than the octonions, since they are non-alternative and do not uphold the Moufang identities. Intriguingly, the real Okubo algebra has $\text{SU}\left(3\right)$ as automorphism group, which is a classical Lie group, while octonions and paraoctonions have an exceptional Lie group of type $\text{G}_{2}$. This is remarkable, given that the projective plane defined over the real Okubo algebra is nevertheless isomorphic and isometric to the octonionic projective plane which is at the very heart of the geometric realisations of all types of exceptional Lie groups. Despite its historical ties with octonionic geometry, our research underscores the real Okubo algebra as the weakest algebraic structure allowing the definition of the compact 16-dimensional Moufang plane.