Algebraic and geometric properties of Fibonacci Cartan numbers

In this paper we developed an algebraic and geometric study of Fibonacci – Cartan numbers and Fibonacci Pauli–Cartan numbers by integrating Fibonacci sequences with the algebraic structure of Cartan numbers in pseudo–Euclidean spaces. Building on the concept of Fibonacci–Cartan numbers, we investigated their fundamental algebraic and geometric properties, including conjugation, character, norm, and causal classification, proving that they are spacelike elements and establishing their geometric interpretation within a Minkowski–type framework. Matrix, polar, and exponential representations were obtained, enabling the derivation of De Moivre-type identities, formulas for powers and roots, and characterizations in terms of the determinant and trace of real matrices. In addition, the powers of Fibonacci–Cartan numbers were represented in terms of Chebyshev polynomials of the first and second kinds, yielding explicit polynomial expressions for their scalar and vector parts. Scalar and vector products were also analyzed, providing a consistent geometric interpretation in a three–dimensional Cartan space. Furthermore, Fibonacci Pauli–Cartan numbers were defined and studied through recurrence relations, Binet formulas, ordinary and exponential generating functions, summation formulas, and real matrix representations. Finally, rotational properties associated with lightlike directions were examined, showing that parabolic rotations degenerate into linear trajectories within the Cartan geometric framework.