Generalized Cauchy-Riemann equations in non-identity bases with application to the algebrizability of vector fields

We complete the work done by James A. Ward in the mid-twentieth century on a system of partial differential equations that defines an algebra {\mathbb{A}} for which this system is the generalized Cauchy-Riemann equations for the derivative introduced by Sheffers at the end of the nineteenth century with respect to {\mathbb{A}}, which is also known as the Lorch derivative with respect to {\mathbb{A}}, and recently simply called {\mathbb{A}} -differentiability. We get a characterization of finite-dimensional algebras, which are associative commutative with unity.