Klein and conformal superspaces, split algebras and spinor orbits

We discuss (Formula presented.) Klein and Klein-conformal superspaces in (Formula presented.) space-time dimensions, realizing them in terms of their functor of points over the split composition algebra (Formula presented.). We exploit the observation that certain split forms of orthogonal groups can be realized in terms of matrix groups over split composition algebras. This leads to a natural interpretation of the sections of the spinor bundle in the critical split dimensions (Formula presented.) and (Formula presented.) as (Formula presented.), (Formula presented.) and (Formula presented.), respectively. Within this approach, we also analyze the non-trivial spinor orbit stratification that is relevant in our construction since it affects the Klein-conformal superspace structure.