A geometric description of the spin-embedding of symplectic dual polar spaces of rank 3

We give a geometrical description of the spin-embedding esp of the symplectic dual polar space Δ≅DW(5,r2) by showing how the natural embedding of W(5,r2) into PG(5,r2) is involved in the Grassmann-embedding egr of Δ. We prove that the map sending every quad of Δ to its nucleus realizes the natural embedding of W(5,r2). Taking the quotient of egr over the space spanned by the nuclei of the quadrics corresponding to the quads of Δ gives an embedding isomorphic to esp.