Anisotropic Spaces of Eight-Dimensional Numbers with Real Components

This paper develops a class of anisotropic spaces of eight-dimensional numbers with real components. Starting from a general bilinear multiplication law, we impose commutativity and associativity and derive the resulting relations among the structure constants. Each eight-dimensional number is represented by an 8×8 matrix, and this representation preserves addition and multiplication. We identify 112 spaces in which the determinant of the representing matrix is a non-negative homogeneous form and, therefore, defines a modulus. The zero divisors are characterized explicitly and shown to form subspaces of dimensions two, four, and six, with prescribed relations among their components. We introduce the spectrum of an eight-dimensional number through the eigenvalues of its matrix representation and establish the principal algebraic properties of the spectral mapping, including its compatibility with addition, multiplication, conjugation, and inversion. A spectral norm is then defined and shown to satisfy the norm axioms. We also identify a distinguished subspace that is closed under multiplication and on which the spectral and Euclidean norms coincide. For elements of unit norm in this subspace, the corresponding matrix representations are orthogonal. The resulting number systems retain associative and commutative multiplication while allowing zero divisors and provide a natural extension of real, complex, and four-dimensional number systems.