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In this paper, we focus on the characterization of Lie algebras of fermionic,bosonic and parastatistic operators of spin particles. We provide a method toconstruct a Lie group structure for the quantum spin particles. We show thesemi-simplicity of a quantum spin particle Lie algebra, and extend the resultsto the Lie group level. Besides, we perform the Iwasawa decomposition of spinparticles at both the Lie algebra and Lie group levels. Finally, we investigatethe coupling of angular momenta of spin half particles, and give a generalconstruction for such a study.

Group-algebraic characterization of spin particles: semi-simplicty, SO(2N) structure and Iwasawa decomposition