Quaternionic construction of the W(F₄) polytopes with their dual polytopes and branching under the subgroups W(B₄) and W(B ₃) × W(A₁)

Four-dimensional F₄ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group W(F₄ ) where the group elements and the vertices of the polytopes are represented by quaternions. Branchings of an arbitrary W(F₄ ) orbit under the Coxeter groups W(B₄) and W(B₃) × W(A₁) have been presented. The role of group theoretical technique and the use of quaternions have been emphasized.