A three-dimensional modification of the Gaussian number field

Abstract

For vectors in $\mathbf{E_3}$ we introduce an associative, commutative and
distributive multiplication. We describe the related algebraical and
geometrical properties, and hint some applications.
Based on properties of hyperbolic (Cliord) complex numbers, we
prove that the resulting algebra $\mathBB{T}$ is an associative algebra over field
which contains subring isomorphic with hyperbolic complex numbers.
Moreover, the algebra $\mathBB{T}$ is isomorphic with direct product $\mathbb{C} \times \mathbb{R}$ so
contains subalgebra isomorphic with the Gaussian complex plane.