Bell gems naturally split dynamics information from $SU(2^{2d}) \rightarrow U(1)^{2^{2d-1}-1} \times SU(2)^{2^{2d-1}}$

Quantum Computation and Quantum Information are continuously growing research areas which are based on nature and resources of quantum mechanics, as superposition and entanglement. In its gate array version, the use of convenient and appropriate gates is essential. But while those proposed gates adopt convenient forms for computational algorithms, in the practice, their design depends on specific quantum systems and stuff being used. Gates design is restricted to properties and limitations of interactions and physical elements being involved, where Quantum Control plays a deep role. Quantum complexity of multipartite systems and their interactions requires a tight control to manipulate their quantum states, either local and non-local ones, but still a reducibility procedure should be addressed. This work shows how a general $2d$-partite two level spin system in $SU(2d)$ could be decomposed in $2^{n-1}$ subsystems on $SU(2)$, letting establish control operations. In particular, it is shown that Bell gems basis is a set of natural states on which decomposition happen naturally under some interaction restrictions. Thus, alternating the direction of local interaction terms in the Hamiltonian, this procedure states a universal exchange semantics on those basis. The structure developed could be understood as a splitting of the $2d$ information channels into $2^{2d-1}$ pairs of $2$ level information subsystems.