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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 overthe split composition algebra $\mathbb{C}_{s}$. We exploit the observation thatcertain split form of orthogonal groups can be realized in terms of matrixgroups over split composition algebras; this leads to a natural interpretationof 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 thenon-trivial spinor orbit stratification that is relevant in our constructionsince it affects the Klein-Conformal superspace structure.

Klein and Conformal Superspaces, Split Algebras and Spinor Orbits