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4-dimensional $F_{4} $ polytopes and their dual polytopes have beenconstructed as the orbits of the Coxeter-Weyl group $W(F_{4})$ where the groupelements and the vertices of the polytopes are represented by quaternions.Branchings of an arbitrary \textbf{$W(F_{4})$} orbit under the Coxeter groups$W(B_{4} $ and $W(B_{3}) \times W(A_{1})$ have been presented. The role ofgroup theoretical technique and the use of quaternions have been emphasized

Quaterionic Construction of the W(F_4) Polytopes with Their Dual Polytopes and Branching under the Subgroups B(B_4) and W(B_3)*W(A_1)