  4-dimensional $F_{4} $ polytopes and their dual polytopes have been
constructed as the orbits of the Coxeter-Weyl group $W(F_{4})$ where the group
elements 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 of
group theoretical technique and the use of quaternions have been emphasized
