Jordan subalgebra - Knowlpedia

Definition

A Jordan subalgebra of a Jordan algebra $J$ over $k$ is a linear subspace $B\subseteq J$ such that

x,y\in B\quad\Longrightarrow\quad x\circ y\in B.

With the restricted product, $B$ is itself a Jordan algebra.

The unit convention

If $J$ is unital, closure under $\circ$ does not require $B$ to contain the ambient unit $e_J$ . A subalgebra can have no unit, or can have a unit $e_B$ different from $e_J$ . Authors who require subalgebras of unital Jordan algebras to contain $e_J$ are using the stronger phrase unital Jordan subalgebra.

For example, the upper-left $2\times2$ Hermitian corner inside $\mathfrak h_3(\mathbb O)$ is closed under the Jordan product. Its unit is $\operatorname{diag}(1,1,0)$ , not the ambient unit $\operatorname{diag}(1,1,1)$ . This is the convention relevant to matrix corner inclusions

\mathfrak h_2(\mathbb C)\subset \mathfrak h_3(\mathbb C)\subset \mathfrak h_3(\mathbb O).

Embeddings

An injective Jordan algebra homomorphism identifies its source with a Jordan subalgebra of its target. For unital algebras, the embedding is unital precisely when it sends the source unit to the target unit. Thus “embedding” alone should not be read as “unital embedding.”

Stabilizers of subalgebras

If a group $G$ acts by Jordan automorphisms on $J$ , the setwise stabilizer of $B$ is

\operatorname{Stab}_G(B)=\{g\in G:g(B)=B\}.

This is different from the pointwise stabilizer, whose elements fix every element of $B$ . In the exceptional Jordan algebra, stabilizers of selected complex and octonionic matrix subalgebras recover important compact Lie groups.

References

Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, 2000. Publisher record.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv:2606.15235.