Compact exceptional Lie group G2

The compact exceptional Lie group $G_2$ is the compact connected simple Lie group of rank $2$ and real dimension $14$ whose root system has Dynkin type $G_2$ . It is both simply connected and centerless. Its real Lie algebra is the compact real form of $\mathfrak g_2$ .

Its distinguished representation is the faithful irreducible action on the seven-dimensional real vector space $\operatorname{Im}\mathbb O$ of imaginary octonions.

Octonion automorphisms

There is a natural isomorphism

G_2\cong\operatorname{Aut}_{\mathbb R\text{-alg}}(\mathbb O).

Every automorphism fixes $1$ , preserves octonion conjugation and the norm, and hence acts orthogonally on $\operatorname{Im}\mathbb O$ . This realizes $G_2$ as a closed subgroup of $SO(7)$ . Equivalently, it is the stabilizer of the alternating three-form

\varphi(x,y,z)=\langle xy,z\rangle

on $\operatorname{Im}\mathbb O$ , with a consistent choice of sign convention.

The use of the division octonions selects the compact form. Automorphisms of the split octonions form the split real group $G_{2(2)}$ , while complexification gives the complex algebraic group of type $G_2$ .

The SU(3) stabilizer

The group $G_2$ acts transitively on the unit sphere in $\operatorname{Im}\mathbb O$ . The stabilizer of a unit imaginary octonion $i$ , equivalently of the complex subalgebra $\mathbb C=\operatorname{span}_{\mathbb R}\{1,i\}$ , is isomorphic to $SU(3)$ . The orthogonal complement of this $\mathbb C$ becomes its defining complex module, yielding

\mathbb O\cong\mathbb C\oplus\mathbb C^3.

This decomposition and its $SU(3)$ -equivariant operations are the starting point for the octonion construction in the exceptional-Jordan-algebra setting.

References

John C. Baez, "The Octonions," Bulletin of the American Mathematical Society 39 (2002), 145--205. DOI.
John Frank Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, 1996, Chapters 3--5. Publisher record.
Robert L. Bryant, "Metrics with exceptional holonomy," Annals of Mathematics 126 (1987), 525--576. DOI.
John C. Baez and Paul Schwahn, The Standard Model Gauge Group from the Exceptional Jordan Algebra, 2026. arXiv:2606.15235.