Spin(8) stabilizer of an Albert-algebra frame

Statement

Let $J=H_3(\mathbb O)$ be the compact real Albert algebra, let $F_4=\operatorname{Aut}(J)$ , and let $(e_1,e_2,e_3)$ be a labelled Jordan frame. Its pointwise stabilizer is

\{g\in F_4:g(e_i)=e_i\text{ for }i=1,2,3\}\cong\mathrm{Spin}(8).

Moreover, $F_4$ acts transitively on labelled Jordan frames.

Triality action

For the standard frame, the decomposition

J=\mathbb R e_1\oplus\mathbb R e_2\oplus\mathbb R e_3 \oplus\xi_{12}(\mathbb O)\oplus\xi_{23}(\mathbb O) \oplus\xi_{31}(\mathbb O)

is invariant under the frame stabilizer. The three eight-dimensional off-diagonal summands carry, in a suitable ordering, the vector, left half-spin, and right half-spin representations of $\mathrm{Spin}(8)$ . Their coupled action is Spin(8) triality.

Pointwise versus setwise stabilizers

The qualifier “pointwise” is essential. If a frame is regarded as an unordered set, its setwise stabilizer is

\mathrm{Spin}(8)\rtimes S_3.

The quotient $S_3$ permutes $e_1,e_2,e_3$ and acts on $\mathrm{Spin}(8)$ through the outer automorphisms associated with triality. Thus the unqualified statement “the setwise stabilizer is $\mathrm{Spin}(8)$ ” is false. Its identity component is the pointwise $\mathrm{Spin}(8)$ .

References

Ichirô Yokota, Exceptional Lie Groups, 2009, Theorem 2.7.1. arXiv:0902.0431.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026, Lemma 5. arXiv:2606.15235.
John F. Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, 1996, Chapters 1–3. Publisher record.