  Extending our earlier study of nonlinear Bogolyubov-Valatin transformations
(canonical transformations for fermions) for one fermionic mode, in the present
paper we perform a thorough study of general (nonlinear) canonical
transformations for two fermionic modes. We find that the Bogolyubov-Valatin
group for n=2 fermionic modes which can be implemented by means of unitary
SU(2^n = 4) transformations is isomorphic to SO(6;R)/Z_2. The investigation
touches on a number of subjects. As a novelty from a mathematical point of
view, we study the structure of nonlinear basis transformations in a Clifford
algebra [specifically, in the Clifford algebra C(0,4)] entailing
(supersymmetric) transformations among multivectors of different grades. A
prominent algebraic role in this context is being played by biparavectors
(linear combinations of products of Dirac matrices, quadriquaternions,
sedenions) and spin bivectors (antisymmetric complex matrices). The studied
biparavectors are equivalent to Eddington's E-numbers and can be understood in
terms of the tensor product of two commuting copies of the division algebra of
quaternions H. From a physical point of view, we present a method to
diagonalize any arbitrary two-fermion Hamiltonian. Relying on Jordan-Wigner
transformations for two-spin-1/2 and single-spin-3/2 systems, we also study
nonlinear spin transformations and the related problem of diagonalizing
arbitrary two-spin-1/2 and single-spin-3/2 Hamiltonians. Finally, from a
calculational point of view, we pay due attention to explicit parametrizations
of SU(4) and SO(6;R) matrices (of respective sizes 4x4 and 6x6) and their
mutual relation.
