SU(3) stabilizer of a complex octonion subalgebra

Statement

Let $i\in\operatorname{Im}(\mathbb O)$ be a unit imaginary octonion, so $i^2=-1$ , and let $\mathbb C_i=\operatorname{span}_{\mathbb R}\{1,i\}$ . The subgroup of

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

that fixes $i$ , equivalently fixes $\mathbb C_i$ pointwise, is isomorphic to $SU(3)$ . Here $G_2$ is the compact exceptional Lie group.

The defining representation

The orthogonal complement $\mathbb C_i^\perp\subset\mathbb O$ is a complex three-dimensional vector space. In the multiplication convention used by the complex-vector model, its complex structure is right multiplication by $i$ . Under an identification $\mathbb C_i^\perp\cong\mathbb C^3$ , the stabilizer acts by the defining representation of $SU(3)$ . This gives the orthogonal splitting

\mathbb O\cong\mathbb C_i\oplus\mathbb C_i^\perp \cong\mathbb C\oplus\mathbb C^3

used in the complex-vector construction of the octonions.

Why the action preserves multiplication

In the $\mathbb C\oplus\mathbb C^3$ model, $SU(3)$ preserves the Hermitian inner product and the complex volume form. It therefore preserves the conjugated cross product, and hence every term in the octonion multiplication. The action

g(a,u)=(a,gu)

is consequently by algebra automorphisms and is trivial on $\mathbb C_i$ .

Homogeneous-space interpretation

The compact group $G_2$ acts transitively on the unit sphere in $\operatorname{Im}(\mathbb O)\cong\mathbb R^7$ . Since the stabilizer of a unit imaginary octonion is $SU(3)$ ,

G_2/SU(3)\cong S^6.

At the Lie-algebra level, the stabilizer has Lie algebra $\mathfrak{su}(3)$ inside the exceptional Lie algebra $\mathfrak g_2$ .

Pointwise versus setwise stabilizer

The $SU(3)$ statement concerns the stabilizer of the chosen element $i$ , or equivalently the pointwise stabilizer of $\mathbb C_i$ . The setwise stabilizer of the two-plane $\mathbb C_i$ has a second component whose elements carry $i$ to $-i$ ; it is an extension of $SU(3)$ by a group of order two. Thus “fixes $i$ ” cannot be replaced by “preserves $\mathbb C_i$ as a set” without changing the group.

References

John C. Baez, “The Octonions,” Bulletin of the American Mathematical Society 39 (2002), 145–205. DOI record. Relevant: §4.1.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv:2606.15235. Relevant: §§2–3.
Ichiro Yokota, Exceptional Lie Groups, Lecture Notes in Mathematics 2369, Springer, 2025. arXiv:0902.0431. Relevant: Theorem 1.9.1.