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We introduce the notion of intrinsic subspaces of linear and affine pairgeometries, which generalizes the one of projective subspaces of projectivespaces. We prove that, when the affine pair geometry is the projective geometryof a Lie algebra introduced in [Bertram-Neeb, J. Alg. 277], such intrinsicsubspaces correspond to inner ideals in the associated Jordan pair, and weinvestigate the case of intrinsic subspaces defined by the Peirce-decompositionwhich 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 theconjecture that geometries of intrinsic subspaces tend to be themselves linearpair geometries.

Inner ideals and intrinsic subspaces of linear pair geometries