  The low-energy and weak-field limit of Dirac equation can be obtained by an
order-by-order block diagonalization approach to any desired order in the
parameter $\pi/mc$ ($\pi$ is the kinetic momentum and $m$ is the mass of the
particle). In the previous work, it has been shown that, up to the order of
$(\pi/mc)^8$, the Dirac-Pauli Hamiltonian in the Foldy-Wouthuysen (FW)
representation may be expressed as a closed form and consistent with the
classical Hamiltonian, which is the sum of the classical relativistic
Hamiltonian for orbital motion and the Thomas-Bargmann-Michel-Telegdi (T-BMT)
Hamiltonian for spin precession. In order to investigate the exact validity of
the correspondence between classical and Dirac-Pauli spinors, it is necessary
to proceed to higher orders. In this paper, we investigate the FW
representation of the Dirac and Dirac-Pauli Hamiltonians by using Kutzelnigg's
diagonalization method. We show that the Kutzelnigg diagonalization method can
be further simplified if nonlinear effects of static and homogeneous
electromagnetic fields are neglected (in the weak-field limit). Up to the order
of $(\pi/mc)^{14}$, we find that the FW transformation for both Dirac and
Dirac-Pauli Hamiltonians is in agreement with the classical Hamiltonian with
the gyromagnetic ratio given by $g=2$ and $g\neq2$ respectively. Furthermore,
with higher-order terms at hand, it is demonstrated that the unitary FW
transformation admits a closed form in the low-energy and weak-field limit.
