Projective completions of Jordan pairs: Part I. The generalized projective geometry of a Lie algebra

We prove that the projective completion (X+,X−) of the Jordan pair corresponding to a 3-graded Lie algebra can be realized inside the space of inner 3-filtrations of in such a way that the remoteness relation on X+×X− corresponds to transversality of flags. This realization is used to give geometric proofs of structure results which will be used in Part II of this work in order to define on X+ and X− the structure of a smooth manifold (in arbitrary dimension and over general base fields or -rings).