# Order 3 elements in G2 and idempotents in symmetric composition algebras

## TL;DR

This paper classifies order three elements in the exceptional group G2 over arbitrary fields, computes their centralizers, and links these to the classification of idempotents in symmetric composition algebras, highlighting differences across characteristics.

## Contribution

It provides a comprehensive classification of order three elements in G2 and their centralizers, and relates these to idempotents in symmetric composition algebras over arbitrary fields.

## Key findings

- Two conjugacy classes of order three elements in G2 over algebraically closed fields (not 3), four in characteristic 3.
- Centralizers in characteristic 3 are not smooth for one class.
- Classification of idempotents in symmetric composition algebras derived from G2 elements.

## Abstract

Order three elements in the exceptional groups of type G2 are classified up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is obtained. Idempotents have played a key role in the study and classification of these algebras. Over an algebraically closed field, there are two conjugacy classes of order three elements in G2 in characteristic not 3 and four of them in characteristic 3. The centralizers in characteristic 3 fail to be smooth for one of these classes.