  We give a Morita equivalence theorem for so-called cyclotomic quotients of
affine Hecke algebras of type B and D, in the spirit of a classical result of
Dipper-Mathas in type A for Ariki-Koike algebras. As a consequence, the
representation theory of affine Hecke algebras of type B and D reduces to the
study of their cyclotomic quotients with eigenvalues in a single orbit under
multiplication by $q^2$ and inversion. The main step in the proof consists in a
decomposition theorem for generalisations of quiver Hecke algebras that
appeared recently in the study of affine Hecke algebras of type B and D. This
theorem reduces the general situation of a disconnected quiver with involution
to a simpler setting. To be able to treat types B and D at the same time we
unify the different definitions of generalisations of quiver Hecke algebra for
type B that exist in the literature.
