We investigate the algebraic structure of the most general neutrino mass Ha
miltonian and place the seesaw mechanism in an algebraic framework. We show
that this Hamiltonian can be written in terms of the generators of an Sp(4
) algebra. The Pauli-Gursey transformation is an SU(2) rotation which is em
bedded in this Sp(4) group. This SU(2) also generates the seesaw mechanism.