  Hom-Lie algebras having non-invertible twist maps in their centroids are
studied. Central extensions of Hom-Lie algebras having these properties are
obtained and shown how the same properties are preserved. Conditions are given
so that the produced central extension has an invariant metric with respect to
its Hom-Lie product making its twist map self-adjoint when the original Hom-Lie
algebra has such a metric. This work is focused on algebras with these
properties and following Benayadi and Makhloufwe call them quadratic Hom-Lie
algebras. It is shown how a quadratic Hom-Lie algebra gives rise to a quadratic
Lie algebra and that the Lie algebra associated to the given Hom-Lie central
extension is a Lie algebra central extension of it. It is also shown that if
the Hom-Lie product is not a Lie product, there exists a non-abelian algebra,
which is in general non-associative too, the commutator of whose product is
precisely the Hom-Lie product of the Hom-Lie central extension. Moreover, the
algebra whose commutator realizes this Hom-Lie product is shown to be simple if
the associated Lie algebra is nilpotent. Non-trivial examples are provided.
