Peirce-one Jordan corner

Core idea

For an idempotent $e$ in a Jordan algebra $J$ , the Peirce-one corner is

J_1(e)=\{x\in J:e\circ x=x\}.

It is a Jordan subalgebra of $J$ , and $e$ is its unit. Notice that the defining equation is $e\circ x=x$ , not $e\circ x=1$ .

Hermitian

3

-by-

3

case

Let $\mathbb K$ be one of the normed real division algebras $\mathbb R,\mathbb C,\mathbb H,\mathbb O$ , and let $J=H_3(\mathbb K)$ . If $\ell\in J$ is an idempotent of matrix trace $2$ , then

J_1(\ell)\cong H_2(\mathbb K)

as a unital Jordan algebra. After an automorphism takes $\ell$ to $E_{11}+E_{22}$ , this corner consists exactly of matrices

\begin{pmatrix} \alpha&x&0\\ x^*&\beta&0\\ 0&0&0 \end{pmatrix}, \qquad \alpha,\beta\in\mathbb R,\quad x\in\mathbb K.

Conversely, every Jordan subalgebra of $H_3(\mathbb K)$ isomorphic to $H_2(\mathbb K)$ is $J_1(\ell)$ for a unique trace-two idempotent $\ell$ : namely, the unit of that subalgebra viewed inside $J$ .

References

John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026, Lemmas 7–10. arXiv:2606.15235.
Nathan Jacobson, Structure and Representations of Jordan Algebras, American Mathematical Society, 1968, Chapter III, §1. Publisher record.