Buildings of type C_n. III. Non-embeddable polar spaces

Abstract

THEOREM. Let K be a division Cayley algebra over a field k, and let n_K : K → k be the quadratic norm form of this algebra. Then, there exists one and, up to isomorphism, only one polar space of rank 3 whose planes are projective planes over K. The cones of this space (7.2.6) are isomorphic to the dual of the polar space associated with the quadratic form K × k⁴ → k defined by

$$ (x_0 ,x_1 ,x_2 ,x_3 ,x_4 ) \mapsto n_K (x_0 ) - x_1 x_3 + x_2 x_4 , $$

(1)

where x₀ ∈ K and x_i ∈ k for i = 1, 2, 3, 4.