Constructive diagonalization of an element X of the Jordan algebra J(3,O^3) by the exceptional group F_4

TL;DR

This paper provides a direct, constructive proof that any element of the exceptional Jordan algebra can be diagonalized by the exceptional Lie group F_4, improving upon previous contradiction-based methods.

Contribution

The authors present a new constructive method for diagonalizing elements of the exceptional Jordan algebra using the group F_4, avoiding contradiction-based proofs.

Findings

Constructive diagonalization method for Jordan algebra elements

Explicit use of F_4 group actions

Simplification over previous proof techniques

Abstract

We know that any element $X$ of the exceptional Jordan algebra $\gJ$ is transformed to a diagonal form by the compact exceptional Lie group $F_4$. However, its proof is used the method which is reduced a contradiction. In this paper, we give a direct and constructive proof.