On orthogonal polar spaces

Let P be a non-degenerate polar space. In [6], we introduced
an intrinsic parameter of P, called the anisotropic gap,
defined as the least upper bound of the lengths of the well-
ordered chains of subspaces of P containing a frame; when
P is orthogonal, we also defined two other parameters of
P, called the elliptic and parabolic gap, both related to the
universal embedding of P. In this paper, assuming that P
is an orthogonal polar space, we prove that the elliptic and
parabolic gaps can be described as intrinsic invariants of P
without directly appealing to the embedding.