Proposition 32 ([Ordered spinor stabilizer]). Let W=W₊ . Under V_K=W⊕W^* , the ordered line stabilizer and the pointwise stabilizer are L_K = (g,t)∈GL(W)×𝔾_m:t²= g , m(g,t)=t^-1 Λg, J_K= (g,1):g∈SL(W) . The characters on λ₊ and λ_- …

The two annihilators in are complementary maximal isotropic spaces, paired perfectly by Q . Identify the second with W^* . The Clifford algebra on W⊕W^* acts on ΛW by wedge and contraction. As in the matrix-unit argument above it is the full matrix algebra on this module. Hence this module is isomorphic to S_K as a Clifford module. The common annihilator of all wedge operators from W is the top line Kω , while the common annihilator of contractions from W^* is the vacuum line K1 . The two original spinor lines therefore correspond to Kω,K1 , in that order. An ordered stabilizer must preserve the two annihilators, so its vector action is diag(g,g^ -T ) for g∈GL(W) . Conversely a lift of such a block action preserves both one-dimensional annihilator spaces. The operator P_g=Λg on ΛW implements this block action by direct conjugation of wedge and contraction. It is in the even Clifford algebra because it preserves parity. For the top reversal pairing, ⟨P_gs,P_gt⟩=(g)⟨s,t⟩. Thus its Clifford reversion norm is P_gP_g= g . Any other Clifford lift of the same vector action differs by a scalar: their quotient commutes with all of the full matrix algebra. A norm-one lift is therefore aP_g with a^2 g=1 . Writing t=a^ -1 proves the asserted algebraic cover and action. The top and vacuum characters are t^ -1 g=t and t^ -1 . Fixing both vectors is exactly t=1 , hence exactly SL(W) . In particular the extra covering element (1,-1) , which negates both spinors, is not in J_K . This is a statement of algebraic groups; it does not assert that every K -rational determinant has a square root in K . The map t↦(diag(t^2,1,…,1),t) is an algebraic section of the character t ; in particular t^2,t^ -2 are distinct nontrivial characters. Now Λ^k(V_K) =_ a+b=k Λ^aW⊗Λ^bW^*. The invariant vectors in a summand are the intertwiners Hom_ SL(W) (Λ^bW,Λ^aW) . Here is the elementary calculation of those intertwiners. In a basis w_1,…,w_6 , the diagonal torus ∏t_i=1 acts on w_I with weight ∏_ i∈I t_i . Two such weights, for I,J⊂1,…,6 , agree on this torus exactly when their indicator vectors differ by a constant multiple of (1,…,1) . Thus either I=J , or (I,J)=( 1,…,6 ,∅) , or the reverse. For 1≤a≤5 , any invariant subspace of Λ^aW is a sum of its one-dimensional torus weight spaces. The elementary unipotents w_i↦w_i+c w_j connect every two a -subsets by successive replacements, so a nonzero invariant subspace contains all the weight spaces. This proves irreducibility. In an endomorphism commuting with the torus the coefficients on these weight spaces are diagonal, and commuting with the same unipotents makes them all equal. Consequently the only invariant blocks are a=b, (a,b)=(6,0),(0,6), and each is one-dimensional. The balanced identity tensor is the degree-two class ∑_i w_i∧w_i^* ; its j -th power is nonzero and spans the balanced block a=b=j . The already constructed nonzero invariant θ_M is a scalar multiple of that class. The other two lines are W, W^* , with ordered-group characters g=t^2 and ( g)^ -1 =t^ -2 . Invariants commute with extension of scalars: they are the kernel of the coaction difference, and field extension is flat. The group J and θ_M are rational. The two determinant lines are exchanged by scalar conjugation and descend to the rational Weil plane. They occupy the exterior summands (a,b)=(6,0),(0,6) in Λ^aW⊗Λ^bW^* , whereas θ_M^3 occupies the (a,b)=(3,3) summand. Thus the sum in degree six is direct. Descent gives , including the vanishing in odd degrees.