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The generalized Fierz identities are addressed in the K\"ahler-Atiyah bundleframework from the perspective of the equations governing constrainedgeneralized Killing spinor fields. We explore the spin geometry in a Riemannian8-manifold composing a warped flux compactification AdS$_3\times M_8$, whosemetric and fluxes preserve one supersymmetry in AdS$_3$. Supersymmetryconditions can be efficiently translated into spinor bilinear covariants, whosealgebraic and differential constraints yield identifying new spinor fieldclasses. Intriguing implications and potential applications are discussed.

New constrained generalized Killing spinor field classes in warped flux compactifications