On orthogonal polar spaces

Let $\mathcal{P}$ be a non-degenerate polar space. In [I. Cardinali, L. Giuzzi, A. Pasini, “The generating rank of a polar grassmannian”, Adv. Geom. 21:4 (2021), 515-539 doi:10.1515/advgeom-2021-0022 (arXiv:1906.10560)] we introduced an intrinsic parameter of $\mathcal{P}$, called the anisotropic gap, defined as the least upper bound of the lengths of the well-ordered chains of subspaces of $\mathcal{P}$ containing a frame; when $\mathcal{P}$ is orthogonal, we also defined two other parameters of $\mathcal{P}$, called the elliptic and parabolic gap, related to the universal embedding of $\mathcal{P}$. In this paper, assuming $\mathcal{P}$ is an orthogonal polar space, we prove that the elliptic and parabolic gaps can be described as intrinsic invariants of $\mathcal{P}$ without making recourse to the embedding.