Topic: Group-Algebraic Characterization of Spin Particles: Semi-Simplicity, $_mathbf{_bSO(2N)}$ Structure and Iwasawa Decomposition. Abstract: In this paper, we focus on the characterization of Lie algebras of fermionic, bosonic and parastatistic operators of spin particles. We provide a method to construct a Lie group structure for the quantum spin particles. We show the semi-simplicity of the Lie algebra for a quantum spin particle, and extend the results to the Lie group level. Besides, we perform the Iwasawa decomposition for spin particles at both the Lie algebra and the Lie group levels. Then, we give a general decomposition for spin particles. Finally, we investigate the coupling of angular momenta of spin half particles, and give a general construction for such a study.