Inductive description of quadratic Hom-Lie algebras with twist maps in the centroid

In this work we give an inductive way to construct quadratic Hom-Lie algebras with twist maps in the centroid. We focus on those Hom-Lie algebras that are not Lie algebras. We prove that the twist map of a Hom-Lie algebra of this type must be nilpotent and the Hom-Lie algebra has trivial center. We also prove that there exists a maximal ideal containing the kernel and the image of the twist map. Then we state an inductive way to construct this type of Hom-Lie algebras -- similar to the double extension procedure for Lie algebras -- and prove that any indecomposable quadratic Hom-Lie algebra with nilpotent twist map in the centroid, which is not a Lie algebra, can be constructed using this type of double extension.