An Algebraic Methodology for Generalizing the Lorentz Transformation Through Quartic Equation Solutions and f(z) gravitational theory

In this paper, we propose a novel methodology to construct the Lagrangian function using the radical solutions of a quartic equation. By examining the generalized coordinates $q$ and velocities $\dot{q}$ expressed through the radical solutions of the quartic equation, we delve into the relationship between the roots of the quartic equation and the Lorentz transformation. Our findings suggest a potential algebraic generalization of the Lorentz transformation, shedding light on the mathematical structures underpinning relativistic physics. This paper presents the construction of an action that resembles an algebraic structure in f(z) gravity. We were able to reconstruct the algebraic structure of the f(z) gravitational theory.