Proposition. The simple type-D operators generate the positive nilradical [fcap-002X]

✍️sourceAGENTDRAFTED

Let \(V=W\oplus W'\) be a complex nondegenerate quadratic space with a chosen split polarization of rank \(\ell \geq 4\). Choose polar-dual bases \(e_i\in W\) and \(e_i'\in W'\) with \(B_Q(e_i,e_j')=\delta _{ij}\), and put \(S=\bigwedge W\). Write \(c_i\) for left creation by \(e_i\), \(a_i\) for contraction by \(e_i'\), \(N_i=c_i a_i\), and \[H_i=N_i-\tfrac 12.\] Thus \(\epsilon _i(H_j)=\delta _{ij}\) fixes the Cartan coordinates, and the occupation signs are those of Theorem [fcap-0025]. We choose \[\Phi ^+=\{\epsilon _i-\epsilon _j,\ \epsilon _i+\epsilon _j:i<j\}\] and the Borel \(\mathfrak b=\mathfrak h\oplus \bigoplus _{\alpha \in \Phi ^+}\mathfrak g_\alpha \). Equivalently, \(\mathfrak b\) stabilizes the flag \(\langle e_1\rangle \subset \langle e_1,e_2\rangle \subset \cdots \subset W\); in the ordered Witt basis \(e_1,\ldots ,e_\ell ,e_\ell ',\ldots ,e_1'\) it is the upper-triangular Borel.

For the simple roots \(\alpha _i=\epsilon _i-\epsilon _{i+1}\) along the chain and \(\alpha _\ell =\epsilon _{\ell -1}+\epsilon _\ell \) at the fork, take \[\begin {aligned} E_i&=c_i a_{i+1},&F_i&=c_{i+1}a_i && (1\leq i<\ell ),\\ E_\ell &=c_{\ell -1}c_\ell ,& F_\ell &=a_\ell a_{\ell -1}.&& \end {aligned}\] The reversed contraction order in \(F_\ell \) fixes the bracket sign. On an increasing exterior word \(u_I\), a chain operator replaces an occupied \(i+1\) by an absent \(i\), or performs the reverse replacement, with coefficient \(1\); otherwise it gives zero. The fork operators create or remove the final pair, again with coefficient \(1\), when the corresponding occupancy condition holds. The CAR then gives \[[E_i,F_i]=h_i,\qquad h_i=N_i-N_{i+1}\ (i<\ell ),\qquad h_\ell =N_{\ell -1}+N_\ell -1.\] Under the bivector map these are the products \(e_i e_{i+1}'\), \(e_{i+1}e_i'\), \(e_{\ell -1}e_\ell \), and \(e_\ell 'e_{\ell -1}'\). Distinct orthogonal factors contribute no hidden scalar term. TauCeti identifies these exact Clifford representatives with the numbered type-D matrix roots and coroots TauCeti.SpinPolarizationData.typeDQuadraticEquiv_rootGenerator TauCeti.SpinPolarizationData.typeDQuadraticEquiv_cartanGenerator.

It remains to pass from the simple roots to the chosen Borel. Put \[X_{ij}=c_i a_j,\qquad Y_{ij}=c_i c_j\qquad (i<j).\] Their Cartan weights are \(\epsilon _i-\epsilon _j\) and \(\epsilon _i+\epsilon _j\). The identities \[\begin {aligned} [X_{ij},X_{jk}]&=X_{ik},\\ [X_{i,\ell -1},Y_{\ell -1,\ell }]&=Y_{i,\ell },\\ [X_{j,\ell },Y_{i,\ell }]&=Y_{ij} \end {aligned}\] for the indicated distinct ordered indices generate every positive \(X_{ij}\) and \(Y_{ij}\) from the simple operators. These are all positive root spaces of \(\mathfrak {so}(V,Q)\), so their span is the full nilradical \(\mathfrak n^+\). In particular, a vector killed by every \(E_i\) is killed by \(\mathfrak n^+\): brackets and linear combinations preserve its annihilator. This final root-space argument supplies the Borel bridge; the formal file presently proves over \(\mathbb Q\) annihilation by the Lie span of the simple positive generators, without identifying that span with a compatible Borel TauCeti.SpinPolarizationData.typeDSpinSerreRepresentation_positiveSimpleRootLieSpan_le_lieAnnihilator.