The twistor space of a real associative algebra: incidence geometry, semisimple classification and slice-regular functions

To every finite-dimensional associative real algebra $A$ we associate its twistor space $S(A)$, the real algebraic set of square roots of $-1$. Left multiplication defines an almost complex structure on $S(A)$, integrable precisely because $A$ is associative. The resulting complex manifold embeds biholomorphically into the Grassmannian of $\mathbb C\otimes A$ by sending $s$ to the $(-i)$-eigenspace of the complexified operator $L_s$.
This realization turns the tautological pairing $\pi:\mathbb C\otimes A\times S\to A$ into an incidence correspondence. For every compact complex subvariety $K\subseteq S$, the associated zero variety $Z_K\subset\mathbb C\otimes A$ is complex analytic by Remmert's theorem, and the pull-back of the incidence variety along a holomorphic lift describes the corresponding zero set. We also give an intrinsic, section-free formulation of the twistor transform of Gentili, Salamon and Stoppato as a holomorphic map into $\mathbb P(\mathcal V\oplus\mathcal V)$ over $S$.
After choosing a Euclidean structure on $A$, we study the compact subvariety $S_0\subset S$ formed by those $s$ for which $L_s$ is orthogonal. For semisimple algebras, the Wedderburn decomposition shows that $S_0$ is a finite union of products of compact Hermitian symmetric spaces, including $O(2m)/U(m)$, $Sp(m)/U(m)$ and complex Grassmannians. In the classical simple cases, we compute explicit equations for its Euclidean zero variety, obtaining isotropic or determinantal cones.
The construction is motivated by slice-regular function theory. For $A=\mathbb H$, it recovers the classical twistor sphere and the Gentili--Salamon--Stoppato transform. Finally, for a compact connected space $S$, we introduce generalized slice-regular functions and extend the maximum modulus principle, the representation formula and a twisted Cauchy--Riemann characterization via holomorphic reparametrizations of $S$.