Polars and antipodal sets for the outer $3$-symmetric space $\mathbb{S}^7 \times \mathbb{S}^7$

Abstract

We determine the polar and the maximal antipodal set $P$ for the outer 3-symmetric space $\mathbb{S}^7 \times \mathbb{S}^7 = \mathrm{Spin}_8/G_2$ where the 3-symmetric structure is given by the triality automorphism $τ$ on $\mathrm{Spin}_8$. It turns out that $P$ has three elements. The 3-symmetric structure extends to a (non-abelian) $S_3$-structure which we also investigate.