Jacobi Identity for the Generalized Cross Product in <img src=image/13441863_01.gif>: A Theoretical and Computational Approach

The classical cross product in three-dimensional Euclidean space <img src=image/13441863_02.gif> satisfies the Jacobi identity, a fundamental property linked to Lie algebra structures. However, this identity's transparent and rigorous generalization to higher dimensions has remained elusive. In this paper, we address this gap by investigating the antisymmetrization of nested cross products involving <img src=image/13441863_03.gif> vectors in <img src=image/13441863_04.gif>. We aim to establish a generalized Jacobi identity valid for all dimensions <img src=image/13441863_05.gif>. Using symbolic computation via the Wolfram Language, we identified consistent algebraic patterns suggesting such a generalization. Building upon these observations, we developed a theoretical framework based on multilinear algebra, permutation groups, and antisymmetrization operators to prove the identity rigorously. Our methodology combines computational experimentation with classical mathematical theory, resulting in a comprehensive analysis. The principal result states that a specific linear combination of cross products, determined by <img src=image/13441863_06.gif>-shuffles, always vanishes in <img src=image/13441863_04.gif>. This provides a natural extension of the classical Jacobi identity, with deep combinatorial and algebraic significance. The main conclusions highlight the strong interplay between computation and abstract algebra in discovering new mathematical properties. This study contributes to the field by expanding the understanding of antisymmetric operations in higher-dimensional spaces and illustrating a robust methodology that bridges symbolic computation and formal proof. Limitations of this work include its current restriction to Euclidean spaces; extensions to more general manifolds or non-Euclidean settings are left for future research. Although the immediate practical and social implications are limited, the theoretical advances presented here lay the groundwork for further exploration in differential geometry, theoretical physics, and computational mathematics. Our findings emphasize the importance of modern computational tools in advancing pure mathematical research.