Jean-Pierre Gazeau
A Structural Rule Behind Octonionic Geometry and Hopf-Type Constructions
I present a unified perspective on two seemingly distinct constructions arising in non-associative algebra and geometric representation theory, both governed by a simple structural rule that we denote by
$
\mathrm{RLLR} \;/\; \mathrm{LRRL}.
$
This rule emerges naturally in the manipulation of Cayley--Dickson algebras and, independently, in the organization of higher Hopf-type fibrations and their octonionic extensions.
This approach relies on a mnemonic and operational framework that clarifies how non-associativity can be consistently controlled through ordered multiplication patterns. In particular, I show that the same combinatorial scheme (RLLR/LRRL) governs:
\begin{itemize}
\item the internal consistency of iterated Cayley--Dickson constructions,
\item the structure of certain octonionic matrix representations,
\item and the coherence of higher Hopf-like geometric mappings.
\end{itemize}
This observation suggests the existence of an underlying structural principle bridging algebraic and geometric aspects of octonionic frameworks. The resulting viewpoint provides a simplified and unified handling of otherwise intricate non-associative computations, and opens the way to further applications in mathematical physics, especially in contexts where exceptional structures and higher symmetries play a role.
