2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66430We study the geometric structure of Lorentzian spin manifolds, which admit imaginary Killing spinors. The discussion is based on the cone construction and a normal form classification of skew-adjoint operators in signature $(2,n-2)$. Derived geometries include Brinkmann spaces, Lorentzian Einstein-Sasaki spaces and certain warped product structures. Exceptional cases with decomposable holonomy of the cone are possible.15 pages, 1 tableDifferential Geometry53C15; 53C50text