Abstract
We study some properties of the nonembeddable polar spaces related to octonion division rings (and related to the index E7,328). More exactly, we classify its subspaces and show that its generating rank is equal to 5, whereas it was always believed to be at least 6. We also study self-projectivities of length 3 in the maximal singular subspaces, and these turn out to be polarities of octonion projective planes. We establish a connection between the conjugacy classes of such polarities and the orbits of the collineation group of the polar space on triples of opposite singular planes. Along the way, we classify all polar spaces for which each self-projectivity of length 3 in a maximal singular subspace is a polarity.
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