Generalized Kosmann derivative and symmetry operators on Dirac structures

We develop a spinorial symmetry framework for metric Dirac structures in H-twisted exact Courant algebroids. A generalized metric equips a Dirac structure with an intrinsic Riemannian Lie-algebroid geometry and identifies it isometrically with the positive subbundle of the generalized tangent bundle. For sections preserving the generalized metric, we prove an exact intertwining between the positive-spinor Kosmann--Dorfman action and the intrinsic Kosmann derivative. The comparison is independent of the auxiliary compatible generalized connection. Within the intrinsic geometry we derive Killing--Yano (KY) and conformal Killing--Yano (CKY) integrability identities, construct Killing-spinor-preserving operators from odd special KY and even special closed CKY forms, and obtain these forms from Hermitian bilinears of Killing spinors with imaginary Killing number. We illustrate the framework on a global Dirac structure over S²× S¹ with nontrivial H-flux class and rank-two anchor. Its spin connection and symmetry operators are explicit, and its Killing-spinor space is a transverse family C^∞(S¹, C²). Finally, intrinsic Killing sections act on Killing spinors by a Lie algebra representation. For a graph Dirac structure over S³, the odd Jacobi identity of the Hermitian-current bracket fails by a phase, and adjoining a single even phase generator yields a Lie superalgebra. These results connect generalized isometries, hidden spinorial symmetries, and flux-sensitive Lie-algebroid geometry.