  The commonly used pseudo-C symmetry $(\nu_{L})^{c}=C\overline{\nu_{L}}^{T}$
is not defined in Lagrangian field theory. In general, there exist two classes
of Majorana fermions; the first is associated with the chiral fermion with the
conventional C and P symmetries, and the second is associated with the Weyl
fermion defined by CP symmetry only and formally characterised by the pseudo-C
symmetry. Taking the seesaw model as an example, it is shown that a generalized
Pauli--G\"{u}rsey (or Bogoliubov-type) canonical transformation converts the
neutrino defined by the Weyl fermion to the neutrino defined by the chiral
fermion and thus to the conventional Majorana fermion, while preserving the
anti-commutation relations. The mixing angles in the weak lepton sector are not
modified by this generalized Pauli--G\"{u}rsey transformation.
