  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.
