Two new classes of hyperplanes of the dual polar space DH(2n-1,4) not arising from the Grassmann embedding

Let Γn(q) denote the geometry of the hyperbolic lines of the symplectic polar space W(2n-1,q),n⩾2. We show that every hyperplane of Γn(q) gives rise to a hyperplane of the Hermitian dual polar space DH(2n-1,q2). In this way we obtain two new classes of hyperplanes of DH(2n-1,4) which do not arise from the Grassmann embedding of DH(2n-1,4).