Klein and Conformal Superspaces, Split Algebras and Spinor Orbits

We discuss $\mathcal{N}=1$ Klein and Klein-Conformal superspaces in $D=(2,2)$ space-time dimensions, realizing them in terms of their functor of points over the split composition algebra ${\mathbb{C}}_s$. We exploit the observation that certain split form of orthogonal groups can be realized in terms of matrix groups over split composition algebras; this leads to a natural interpretation of the the sections of the spinor bundle in the critical split dimensions $D=4$, $6$ and $10$ as ${\mathbb{C}}_s^2$, ${\mathbb{H}}_s^2$ and ${\mathbb{O}}_s^2$, 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.