The Doublet Tower of Cl(4,2): Universal Nilpotency, Projector Identities, and the sl(8,R) Parabolic from Conformal Null Vectors

The conformal geometric algebra Cl(4,2) R(8) Cl ( 4 , 2 ) ≅ R ( 8 ) carries a natural Z₂ Z 2 -grading defined by the dilatation bivector D=e₄e₀ D = e 4 e 0 , which satisfies D²=1 D 2 = 1 and splits the eight-dimensional spinor module into two four-dimensional eigenspaces S=S₊⊕ S₋ S = S + ⊕ S - . We prove that the adD ad D -eigenvectors of Cl(4,2) Cl ( 4 , 2 ) , those elements with eigenvalue ± 1 ± 1 under the Lie bracket [D,· ] [ D , · ] , are exhausted by 16 doublets of the form T^± J=n_• eJ T J ± = n ∙ e J , where n_• ∈ ∞,nₒ\ n ∙ ∈ { n ∞ , n o } is a null vector and eJ