We construct a noncommutative algebra C(2) that is a subalgebra of the Pauli
matrices of M(2;C), and investigate the properties of solutions with values
in C(2) of the inhomogeneous Cauchy-Riemann system of partial differential
equations with coefficients in the associated Pauli matrices. In addition, we
construct a commutative subalgebra C(4) of M(4;C), obtain some properties of
biregular functions with values in C(2) on in C2 x C2, define a J-regular
function of four complex variables with values in C(4), and examine some
properties of J-regular functions of partial differential equations.
Some partial differential equations in Clifford analysis