Inner ideals in Lie algebras and spherical buildings

Inner ideals of real simple Lie algebras

“…In the case that L is finite-dimensional, simple, and generated (as an algebra) by pure extremal elements, if the set of lines is non-empty, then the extremal geometry is a root shadow space (for instance see [CI07]). Very recently [Coh21] extends Faulkner's results showing that the correspondence between inner ideals of the Lie algebra of a simple algebraic group and shadows on the set of long root groups of the building associated with the algebraic group holds for fields of characteristic different from two too.…”

Inner Ideals of Real Simple Lie Algebras

“…In the case that L is finite-dimensional, simple and generated (as an algebra) by pure extremal elements, if the set of lines is non-empty, then the extremal geometry is a root shadow space (for instance see [11]). Very recently, [13] extends Faulkner's results, showing that the correspondence between inner ideals of the Lie algebra of a simple algebraic group and shadows on the set of long root groups of the building associated with the algebraic group holds for fields of characteristic different from two too.…”

“…The same arguments as above yield Lemma 5. 13 The vector space of dimension 7 given by g 2 = U is a point space of g.…”