Octonion hermitian quadrangles (Q1281143)

A hermitian quadrangle \(H_nF\) is a polar space (of rank~2) defined by a hermitian sesquilinear form on~\(F^{n+1}\). This works nicely over skew fields like \(\mathbb R\), \(\mathbb C\) or~\(\mathbb H\), but the lack of associativity presents difficulties if one wants to extend the definition to the alternative algebra~\(\mathbb O\) of Cayley numbers. The author gives a different but still rather simple definition of \(H_n\mathbb F\) for \(\mathbb F\in\{\mathbb R,\mathbb C,\mathbb H\}\) which works over~\(\mathbb O\), as well, yielding a sequence \((H_n\mathbb O)_{n\geq 3}\) of quadrangles. The construction as well as the proof of the incidence properties are purely algebraic, and work over every algebraically closed ground field. It turns out that the quadrangles \(H_n\mathbb O\) are just those that were constructed by \textit{G.~Thorbergsson} [Duke Math. J. 67, 627-632 (1992; Zbl 0765.51015)] from isoparametric hypersurfaces found by \textit{D.~Ferus, H.~Karcher} and \textit{H.-F.~Münzner} [Math. Z. 177, 479-502 (1981; Zbl 0452.53032)]. The quadrangle \(H_3\mathbb O\) is singled out by the property that its group of automorphisms acts transitively on the set of lines. Indeed, this quadrangle can be reconstructed from the action of a suitable compact subgroup.