  We introduce the notion of intrinsic subspaces of linear and affine pair
geometries, which generalizes the one of projective subspaces of projective
spaces. We prove that, when the affine pair geometry is the projective geometry
of a Lie algebra introduced in [Bertram-Neeb, J. Alg. 277], such intrinsic
subspaces correspond to inner ideals in the associated Jordan pair, and we
investigate the case of intrinsic subspaces defined by the Peirce-decomposition
which is related to 5-gradings of the projective Lie algebra. These examples,
as well as the examples of general and Lagrangian flag geometries, lead to the
conjecture that geometries of intrinsic subspaces tend to be themselves linear
pair geometries.
