SO(2n+1) in an SO(2n-3)?SU(2)?SU(2) basis. I. Reduction of the symmetric representations

The branching rule for the reduction of symmetric irreducible unitary representations (IUR) of the simple Lie group SO(2n+1) into IUR of its maximal subgroup SO(2n-3)(X)SU(2)(X)SU(2) is established for all n>or=3. After the particular case n=3 is analysed in detail, a general proof is presented which is valid for all n>or=3. All branching rules (n=3,4,...) can be summed up in one formula. Also, a dimension verification is carried out. The generators of SO(2n+1) not belonging to the semi-simple subgroup can be combined into a mixed tensor-spinor representation with respect to the simple groups which occur in the direct product. The precise nature of that representation is indicated and discussed.