arXiv:2611.14547v1 Announce Type: new Abstract: This work investigates the involutive, adjoint, and spectral structures arising in the extended Alpha Group. Starting from the four-dimensional Alpha Group transformation matrix, whose angular depen- dence is governed by coupled tan {\theta} and cot {\theta} modes, an iterative commutator procedure generates a sixteen-dimensional real Lie algebra. Within this algebra, the operators B5 and B6 are identified as structurally distinguished elements and analyzed through their adjoint actions. The operator B6 is veri- fied to be involutive and induces an 8 + 8 decomposition of the Lie algebra, while the adjoint action of B5 defines an eight-dimensional invariant mixed sector through a polynomial kernel. The restricted adjoint operators preserve this sector, and their coupled action generates an idempotent operator Q, satisfying Q2 = Q, revealing an invariant projective structure. The spectral analysis further identifies characteristic- polynomial factorizations associated with B5 , B6 , and their compositions, including golden-ratio relations among distinct generators of the extended algebra. These results establish structural connections between the original modal representation, Lie-algebraic involution, adjoint dynamics, projective organization, and the spectral structure of the extended Alpha Group.
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