Point spaces in exceptional quadratic Jordan algebras (Q1239246)

An element \(b\) of a simple exceptional finite dimensional quadratic Jordan algebra \(J\) is said to be of rank 1 if \(JU_b = \Phi b\), where \(\Phi\) is the base field. A subspace of \(J\) consisting entirely of elements of rank 1 is a point space. These are of interest in ideal theory and in geometry. In this paper point spaces of \(J\) are classified up to action by \(\Aut J\), the automorphism group of \(J\). \textit{K. McCrimmon} [Trans. Am. Math. Soc. 159, 445--468 (1971; Zbl 0224.17011)] has classified point spaces of \(J\) up to action by \(\text{Str }J\), the structure group of \(J\). Our methods depend on this classification and are mostly computational.