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.