Let
$\Delta $ be the dual of a
thick polar space
$\Pi $ of rank
$4$. The points, lines, quads,
and hexes of
$\Delta $ correspond with
the singular
$3$spaces, planes,
lines, respectively points of
$\Pi $.
Pralle and Shpectorov have investigated
ovoidal hyperplanes of
$\Delta $
which intersect every hex in the extension of an ovoid of a quad. With every
ovoidal hyperplane there corresponds a unique generalized quadrangle
$\Gamma $. In the finite
case,
$\Gamma $ has been
classified combinatorially, and it has been shown that only the symplectic and elliptic dual polar
spaces
$DS{p}_{8}\left(q\right)$ and
$D{O}_{10}^{}\left(q\right)$ of Witt index
$4$ have ovoidal
hyperplanes. For
$DS{p}_{8}\left(\mathbb{K}\right)$ over
an arbitrary field
$\mathbb{K}$,
it holds
$\Gamma \cong S{p}_{4}\left(\mathbb{H}\right)$ for
some field
$\mathbb{H}$.
In this paper, we construct an embedding projective space for the generalized
quadrangle
$\Gamma $
arising from an ovoidal hyperplane of the orthogonal dual polar space
$D{O}_{10}^{}\left(\mathbb{K}\right)$ for a field
$\mathbb{K}$. Assuming
$char\left(\mathbb{K}\right)\ne 2$ when
$\mathbb{K}$ is infinite, we
prove that
$\Gamma $
is a hermitian generalized quadrangle over some division ring
$\mathbb{H}$.
Moreover we show that an ovoidal hyperplane
$H$ arises from the
universal embedding of
$\Delta $,
if the ovoids
$Q\cap H$ of
all ovoidal quads
$Q$
are classical. This condition is satisfied for the finite dual polar spaces
$DS{p}_{8}\left(q\right)$ and
$D{O}_{10}^{}\left(q\right)$ by
Pralle and Shpectorov.
