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19.92 (2018)

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Let $A$ be the algebra of $3 \times 3$ skew-Hermitian matrices over the real octonions, where multiplication is given by bracket product. Determine $\text{Aut}(A)$.

The question might be of interest for octonion algebras over a different base field or ring. The question was inspired by an observation of John Faulkner concerning a possible connection between $A$ or a related algebra and the Dwyer–Wilkerson 2-compact group $\text{BDI}(4)$.

Contributor: R. Solomon

Progress

*It is determined: $\text{Aut}(A) \cong G_2 \times \text{SO}(3)$, which follows from the result about derivations in (H. Petyt, Commun. Algebra, 47, no. 10 (2019), 4216–4223).

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