Octonionic Planes and Real Forms of G2, F4 and E6

TL;DR

This paper explores the structure of octonionic planes, their relation to exceptional Jordan algebras, and classifies all such planes as symmetric spaces, connecting them with real forms of G2, F4, and E6 groups.

Contribution

It introduces a new approach to octonionic planes using Veronese vectors and classifies all octonionic and split-octonionic planes as symmetric spaces.

Findings

Veronese vectors correspond to rank-one elements in the Jordan algebra.

All real forms of G2, F4, and E6 are recovered as groups of motions.

Complete classification of octonionic and split-octonionic planes as symmetric spaces.

Abstract

In this work we present a useful way to introduce the octonionic projective and hyperbolic plane through the use of Veronese vectors. Then we focus on their relation with the exceptional Jordan algebra and show that the Veronese vectors are the rank-one elements of the algebra. We then study groups of motions over the octonionic plane recovering all real forms of G2, F4and E6 groups and finally give a classification of all octonionic and split-octonionic planes as symmetric spaces.