Remark. Horizons beyond the exterior model
[meinrenken2013clifford, Section 3.2.4, Theorem 3.3, pp. 59--60];
[kostant1997clifford, Section 2.4, Theorem 8, pp. 286--287];
[lawson2016spin, I.2, pp. 13--19] [fcap-0017]
Remark. Horizons beyond the exterior model [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, pp. 59--60]; [kostant1997clifford, Section 2.4, Theorem 8, pp. 286--287]; [lawson2016spin, I.2, pp. 13--19] [fcap-0017]
The exterior model now has two distinct faithful-range results. In even split dimension, Theorem [fcap-002V] constructs all matrix units, identifies the two even endomorphism blocks, and records irreducibility after restriction to Spin under its separably closed, positive-rank hypotheses. In odd complex dimension, Proposition [fcap-002W] shows that either full Clifford action selects one simple summand, while its restriction to the even algebra is faithful and its actual Spin action exchanges exterior parity in positive rank. Thus exterior parity is not a decomposition into odd-dimensional Spin submodules.
The occupation lines of Theorem [fcap-0025] have also been joined to root and highest-weight theory. The type-D calculation in Proposition [fcap-002X] and Theorem [fcap-002Y] fixes a Borel, proves positive-nilradical generation, and identifies the two fork modules. The type-B calculation in Theorem [fcap-002Z] adds the short root and its irreducible spin module, while Proposition [fcap-0030] derives the finite type-B and type-D characters on the actual Spin torus. The selected TauCeti declarations formalize the Clifford root actions, annihilation identities, and weight vectors. The complex Borel, scalar-extension, irreducibility, torus-descent, and character conclusions are the mathematical arguments in these notes, not one packaged formal theorem.
The low-dimensional complex group comparisons are explicit in Proposition [fcap-0031], Proposition [fcap-0032], Proposition [fcap-0033], and Proposition [fcap-0034]: they identify the actual Spin carriers in dimensions three through six and the stated standard or half-spin modules. These group inverses and module intertwiners are separate from the real algebra classification in Remark [fcap-0016] and its signature-\((1,3)\) instance in Example [fcap-0026]. The concrete real differential comparison in Proposition [fcap-0037] realizes Spin as a closed Lie subgroup, identifies its tangent algebra with the quadratic Clifford lift, and differentiates the vector and finite-dimensional Clifford-module actions; Example [fcap-0038] fixes the planar half-angle normalization. The selected TauCeti declarations formalize the algebraic lift and generator action, while the Lie-group, exponential, and tangent conclusions are the source-grounded arguments in those cards. The based-root calculation in Proposition [fcap-003B] fixes the permutation of the three eight-dimensional highest-weight classes. The actual compact-group lifts and kernel calculation in Theorem [fcap-003C] prove \(\operatorname {Out}_{\mathrm {Lie}}(\operatorname {Spin}_8)\simeq S_3\), while Proposition [fcap-003D] constructs the invariant vector--half-spin trilinear and proves its multiplicity-one uniqueness. The pinned tripled-carrier automorphism is formal context on its stated carrier; it is not used as a packaged theorem about the actual Spin group.