F4 is one of the six exceptional groups of Coxeter E6, E7, E8, F4, H3 and H4. For finite Coxeter groups create also three one-parameter families of increasing rank An, Bn, Dn and one-parameter family of dimension two I2 (p). Coxeter classified them in 1935 with what we call now Coxeter-Dynkin diagrams.
F4 is described in this context as: F4 = [3,4,3]

Each node represents a mirror and he label attached to the branch encodes the dihedral angle order between two mirrors. The unlabeled branches implicitly represent the order 3.
The Bruhat order is a partial order on the group W where (W,S) is a Coxeter system with generators S.
The reduced word for an element w of W is a minimal length expression of w as a product of elements of S and the length l(w)of w is the length of a reduced word.
The Bruhat order is defined by u ≤ v if some substring of some reduced word for v is a reduced word for u.
F4 is a solvable group of order 1152.
The study Dyer’s conjecture on Coxeter groups, we compute all the intervals of length 4 of the Bruhat order of the Coxeter F4 and we find 24 up to isomorphy. At that point, it is possible to see a relation with the 24-cell with its regular net.