Albert algebras and construction of the finite simple groups $F_4(q)$, $E_6(q)$ and ${}^2E_6(q)$ and their generic covers

TL;DR

This paper presents a uniform construction method for certain finite simple groups, specifically $E_6(q)$, $F_4(q)$, and ${}^2E_6(q)$, avoiding special cases for characteristics 2 and 3 and not relying on quadratic Jordan algebras.

Contribution

It offers a coherent, self-contained construction approach for these groups, synthesizing scattered historical research into a unified framework.

Findings

Provides a uniform construction for $E_6(q)$, $F_4(q)$, and ${}^2E_6(q)$

Avoids special treatment for characteristics 2 and 3

Eliminates the need for quadratic Jordan algebras

Abstract

We give a uniform construction of the finite simple groups $E_6(q)$, $F_4(q)$ and ${}^2E_6(q)$, which does not require any special treatment for characteristics 2 or 3, and in particular avoids any mention of quadratic Jordan algebras. Although almost all the ingredients can already be found scattered through research papers spanning more than a century, a coherent, sef-contained, account is hard to find in the literature.