  We develop a Clifford algebra approach for 3D Ising model. By utilizing some
mathematical facts of the direct product of matrices and their trace, we expand
the dimension of the transfer matrices V of the 3D Ising system by adding unit
matrices I (with compensation of a factor) and adjusting their sequence, which
do not change the trace of the transfer matrices V (Theorem I: Trace Invariance
Theorem). It allows us to perform a linearization process on
sub-transfer-matrices (Theorem II: Linearization Theorem). It is found that
locally for each site j, the internal factor Wj in the transfer matrices can be
treated as a boundary factor, which can be dealt with by a procedure similar to
the Onsager-Kaufman approach for the boundary factor U in the 2D Ising model.
This linearization process splits each sub-transfer matrix into 2n sub-spaces
(and the whole system into 2nl sub-spaces). Furthermore, a local transformation
is employed on each of the sub-transfer matrices (Theorem III: Local
Transformation Theorem). The local transformation trivializes the non-trivial
topological structure, while it generalizes the topological phases on the
eigenvectors. This is induced by a gauge transformation in the Ising gauge
lattice that is dual to the original 3D Ising model. The non-commutation of
operators during the processes of linearization and local transformation can be
dealt with to be commutative in the framework of the Jordan-von Neumann-Wigner
procedure, in which the multiplication in Jordan algebras is applied instead of
the usual matrix multiplication AB (Theorem IV: Commutation Theorem).
