Group theoretical analysis of 600-cell and 120-cell 4D polytopes with quaternions

600-cell {3, 3, 5} and 120-cell {5, 3, 3} four-dimensional dual polytopes relevant to quasicrystallography have been studied with the quaternionic representation of the Coxeter group W(H₄). The maximal subgroups W(SU(5)):Z₂ and W(H₃) × Z₂ of W(H ₄) play important roles in the analysis of cell structures of the dual polytopes. In particular, the Weyl-Coxeter group W(SU(4)) is used to determine the tetrahedral cells of the polytope {3, 3, 5}, and the Coxeter group W(H₃) is used for the dodecahedral cells of {5, 3, 3}. Using the Lie algebraic techniques in terms of quaternions, we explicitly construct cell structures forming the vertices of the 4D polytopes.