Exceptional Jordan algebra

Definition

The exceptional Jordan algebra, or real Albert algebra, is

\mathfrak h_3(\mathbb O) =\{X\in M_3(\mathbb O):X^*=X\}, \qquad X\circ Y=\frac12(XY+YX).

It is a 27-dimensional simple Euclidean Jordan algebra with unit $I_3$ , and it is not special.

Coordinates and dimension

An element has the form

\begin{pmatrix} \alpha&z&\bar y\\ \bar z&\beta&x\\ y&\bar x&\gamma \end{pmatrix}, \qquad \alpha,\beta,\gamma\in\mathbb R,\quad x,y,z\in\mathbb O.

The three real diagonal coordinates and three octonionic off-diagonal entries give dimension $3+3\cdot8=27$ . Although octonionic matrix multiplication is not associative, a product of two matrices is unambiguous, and alternativity is sufficient for the degree-three symmetrized product to obey the Jordan identity.

Rank and cubic norm

The Albert algebra has rank $3$ . Its Jordan trace is $\alpha+\beta+\gamma$ , and it has a cubic Jordan determinant. Its characteristic polynomial is

t^3-\operatorname{tr}_J(X)t^2+s(X)t-\det_J(X).

Exceptional character and symmetry

It is exceptional because it cannot be embedded in $A^+$ for any associative algebra $A$ . Over $\mathbb R$ it is the unique exceptional simple Euclidean Jordan algebra. Other fields and other real forms admit Albert algebras with different behavior, so this knowl concerns the compact Euclidean real form.

Its automorphism group is the compact real form of $F_4$ , of dimension $52$ . The larger determinant-preserving structure group is the real form $E_{6(-26)}$ .

References

Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, 2000. Publisher record.
Pascual Jordan, John von Neumann, and Eugene Wigner, “On an Algebraic Generalization of the Quantum Mechanical Formalism,” Annals of Mathematics 35 (1934), 29–64. JSTOR record.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv:2606.15235.