A three dimensional modification of the Gaussian number field

For vectors in $\mathbb{E}_3$ we introduce an associative, commutative and distributive multiplication. We describe the related algebraic and geometrical properties, and hint some applications. Based on properties of hyperbolic (Clifford) complex numbers, we prove that the resulting algebra $\mathbb{T}$ is an associative algebra over a field and contains a subring isomorphic to hyperbolic complex numbers. Moreover, the algebra $\mathbb{T}$ is isomorphic to direct product $\mathbb{C} \times \mathbb{R}$, and so it contains a subalgebra isomorphic to the Gaussian complex plane.

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