The properties of the branching and seam spaces of conical intersections in a molecule with an odd number of electrons are explored for the general case, where the molecule has no spatial symmetry and the Hamiltonian explicitly includes the spin-orbit interaction. A realization of the homomorphism connecting the symplectic group of order 4, Sp(4), and the group of proper rotations in five dimensions SO(5) is used to find an orthogonal representation of the branching space that preserves the standard form of the electronic Hamiltonian near a conical intersection. An invariant property of the branching space is also identified. These findings extend previous results for the nonrelativistic Hamiltonian and the relativistic Hamiltonian with C-s symmetry. A model Hamiltonian representing a tetra-atomic molecule with three coupled doublet electronic states is used to demonstrate the efficacy of the approach and illustrate possible seam loci. The seam of conical intersection is shown to have two distinct branches, one bounded and one infinite in extent. The branching spaces of these seams are characterized. (C) 2003 American Institute of Physics.