Automorphism group of a Jordan algebra

Definition

For a Jordan algebra $J$ , its automorphism group is

\operatorname{Aut}(J)= \{g\in GL(J):g(x\circ y)=g(x)\circ g(y) \text{ for all }x,y\in J\}.

The group operation is composition.

Units and preserved structure

If $J$ is unital, every automorphism fixes its unit: $g(e)$ acts as a unit on the surjective image $g(J)=J$ , and the unit is unique. Automorphisms also preserve idempotents, Jordan frames, rank, and the coefficients of the Jordan characteristic polynomial.

For a Euclidean Jordan algebra, automorphisms preserve the canonical trace form. Consequently $\operatorname{Aut}(J)$ is a compact Lie group. Its Lie algebra is the derivation algebra

\operatorname{Der}(J)= \{D:D(x\circ y)=D(x)\circ y+x\circ D(y)\}.

Examples

For a spin factor $J(V)$ , $\operatorname{Aut}(J(V))\cong O(V)$ .
The identity component of $\operatorname{Aut}(\mathfrak h_n(\mathbb C))$ is $PU(n)$ , acting by unitary conjugation. For $n\geq3$ , complex conjugation supplies another component.
$\operatorname{Aut}(\mathfrak h_3(\mathbb O))$ is the compact exceptional Lie group $F_4$ .

Automorphisms versus stabilizers

If $B\subset J$ is a Jordan subalgebra, restriction gives a homomorphism

\operatorname{Stab}_{\operatorname{Aut}(J)}(B)\longrightarrow \operatorname{Aut}(B).

It need be neither injective nor surjective. Thus the automorphism group of an abstract subalgebra should not be identified with its stabilizer inside a larger algebra.

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Publisher record.
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.