# Constructions for octonion and exceptional Jordan algebras

## Author

**L. J. Rylands** and **D. E. Taylor**

## Status

Research Report 2000-2

Date: 10 January 2000

## Abstract

In this note we reverse the usual process of constructing the Lie
algebras of types *G*_{2} and *F*_{4}
as algebras of derivations of
the split octonions and the exceptional Jordan algebra and instead
begin with their Dynkin diagrams and then construct the algebras
together with an action of the Lie algebras and associated
Chevalley groups. This is shown to be a variation on a general
construction of all standard modules for simple Lie algebras and
it is well suited for use in computational algebra systems. All
the structure constants which occur are integral and hence the
construction specialises to all fields, without restriction on the
characteristic, avoiding the usual problems with characteristics 2
and 3.

## Key phrases

Exceptional Lie algebras. Jordan algebras. Octonions. Derivation algebras

## AMS Subject Classification (1991)

Primary: 17B25

Secondary: 17C40,17D05

## Content

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