Compact exceptional Lie group F4

The compact exceptional Lie group $F_4$ is the compact connected simple Lie group of rank $4$ and real dimension $52$ whose root system has Dynkin type $F_4$ . It is both simply connected and centerless. Its real Lie algebra is the compact real form of $\mathfrak f_4$ .

Its distinguished $26$ -dimensional real representation is the action on the trace-zero part of the exceptional Euclidean Jordan algebra.

Automorphisms of the Albert algebra

Let

H_3(\mathbb O)=\{x\in M_3(\mathbb O):x=x^*\}

with Jordan product $x\circ y=(xy+yx)/2$ . Although octonionic matrix multiplication is not associative, this symmetrized product makes $H_3(\mathbb O)$ a $27$ -dimensional real Jordan algebra. There is a natural isomorphism

F_4\cong\operatorname{Aut}(H_3(\mathbb O)).

The group fixes the identity matrix, so its $27$ -dimensional action decomposes as the trivial line plus the irreducible $26$ -dimensional trace-zero module.

This realization specifies the compact group because $\mathbb O$ is the real division-octonion algebra and $H_3(\mathbb O)$ is Euclidean. Using split octonions gives a noncompact real form instead. Complexifying the Lie algebra gives $\mathfrak f_4$ , not a compact group.

Stabilizers and the Standard Model gauge group

In the exceptional-Jordan-algebra construction, $F_4$ acts on Jordan subalgebras of $H_3(\mathbb O)$ . For nested subalgebras

X\cong H_2(\mathbb C)\subset B\cong H_3(\mathbb C),

one obtains

\operatorname{Stab}_{F_4}(X)\cap\operatorname{Stab}_{F_4}(B)^0 \cong S(U(2)\times U(3)).

The identity-component superscript is essential: the full stabilizer of $B$ has an additional disconnected component induced by complex conjugation.

References

John Frank Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, 1996, Chapters 5--6. Publisher record.
Kevin McCrimmon, A Taste of Jordan Algebras, Springer, 2004, Chapter 6. Publisher record.
John C. Baez, "The Octonions," Bulletin of the American Mathematical Society 39 (2002), 145--205. DOI.
John C. Baez and Paul Schwahn, The Standard Model Gauge Group from the Exceptional Jordan Algebra, 2026. arXiv:2606.15235.