Tsuifukusosu (Tsui-Complex Numbers):A Structural Extension of the Complex Field and A Framework for Commutativization of 4X4 Matrices

Complex numbers have long served as a foundational algebraic structure for describing two-dimensional rotations. However, their internal structure is inherently single-layered, relying on a single phase and a single rotational degree of freedom. This limitation prevents complex numbers from directly expressing higher-order symmetries or multi-layered rotational structures. In this work, we introduce a new algebraic system called \Tsuif, constructed by duplicating the internal rotational structure of the complex field. Tsuifukusosu exhibits algebraic properties distinct from complex numbers, bicomplex numbers, and quaternions. A central result of this paper is that Tsuifukusosu admits a natural representation as $4\times 4$ real matrices, within which a remarkable phenomenon occurs: a nontrivial commutative subalgebra emerges naturally. Since commutativity among general $4\times 4$ matrices is measure-theoretically rare, this natural appearance of commutativity is highly nontrivial. Furthermore, we propose a new framework for transforming noncommutative matrices into commutative ones, called \emph{commutativization}, based on the structural properties of Tsuifukusosu. This framework offers new perspectives for linear algebra and potential applications in quantum mechanics, where noncommutative operators play a central role. In this work, we introduce a new algebraic system called \Tsuif, constructed by duplicating the internal rotational structure of the complex field. **Here “duplication” refers not to introducing two imaginary units, but to a structural duplication realized through its canonical $4\times 4$ representation.** Tsuifukusosu exhibits algebraic properties distinct from complex numbers, bicomplex numbers, and quaternions.