  We establish branching rules between some Iwahori-Hecke algebra of type B and
their subalgebras which are defined as fixed subalgebras by involutions
including Goldman involution. The Iwahori-Hecke algebra of type D is one of
such fixed subalgebras. We also obtain branching rules between those fixed
subalgebras and their intersection subalgebra. We determine basic sets of
irreducible representations of those fixed subalgebras and their intersection
subalgebra by making use of generalized Clifford theory.
