  We propose a new approach to the spinor-spinor R-matrix with orthogonal and
symplectic symmetry. Based on this approach and the fusion method we relate the
spinor-vector and vector-vector monodromy matrices for quantum spin chains. We
consider the explicit spinor R matrices of low rank orthogonal algebras and the
corresponding RTT algebras. Coincidences with fundamental R matrices allow to
relate the Algebraic Bethe Ansatz for spinor and vector monodromy matrices.
