# Octonions, Albert vectors and the group $\mathrm{E}_6(F)$

## TL;DR

This paper introduces a Lie-theory-free, uniform method for constructing and analyzing groups of type E6 over arbitrary fields, simplifying generator description and subgroup analysis, and enabling easier computation of group order in finite cases.

## Contribution

It provides a novel, uniform construction of E6 groups over any field without Lie theory, with explicit generators and subgroup structure insights.

## Key findings

- Simplified description of E6 group generators
- Explicit subgroup structure analysis
- Efficient computation of group order in finite cases

## Abstract

We present a uniform approach to the construction of the groups of type $\mathrm{E}_6$ over arbitrary fields without using Lie theory. This gives a simple description of the group generators and some of the subgroup structure. In the finite case our approach also permits relatively straightforward computation of the group order.