This paper maps a clear link between symmetry operations of anti-de Sitter space (AdS) and a family of gravity solutions in four dimensions. The authors show that, just as three-dimensional AdS solutions like the BTZ black hole, deficit angles and naked singularities come from identifying points using elements of the so(2,2) symmetry algebra, many four-dimensional AdS gravity solutions arise from elements of the so(2,3) algebra. Each type of symmetry element gives a distinct “double copy” gravity solution. The authors classify all such elements and generate the corresponding solutions, which include AdS black holes, black branes and other spacetimes.

The work builds on the classical double copy idea. In that idea, a gravity solution can be assembled from simpler gauge-theory building blocks. One common way to see this is the Kerr–Schild form of a metric, where the full metric equals a background AdS metric plus a term built from a scalar times the square of a special null vector. In that representation the extra piece can be read as coming from a gauge field and a scalar. The paper constructs these Kerr–Schild double copies and a related Weyl double copy. It gives algebraic, coordinate-independent formulas for the gauge field, for the Kerr–Schild vector, and for the scalar “zeroth copy,” by using a kind of Penrose-type transform adapted to AdS.

Technically, the authors sort the so(2,3) elements by their adjoint orbits. Different orbits produce different double copies. The leftover symmetries of each generated solution are precisely the elements that commute with the chosen isometry; mathematically, this set is the centralizer of the isometry. That centralizer lets them pick canonical coordinates tied to the Abelian part of the symmetry. The paper also notes that two Casimir invariants of so(2,3) — algebraic numbers built from the isometry element — appear explicitly in the resulting metrics. The authors provide organized results (their Tables 1 and 2) and give a Penrose-type integrating flow that connects the isometry data to the gravity fields.