  We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean
space and present the group elements in terms of quaternions. It turns out that
it is a maximal subgroup of both the rank-4 Coxeter groups W(F4) and W(H4)
implying that it is a group relevant to the crystallography as well as
quasicrystallographic structures in 4-dimensions. First we review the
pyritohedral symmetry in 3 dimensional Euclidean space which is a maximal
subgroup both in the Coxeter-Weyl groups W(B3)=Aut(D3) and W(H3). The related
polyhedra in 3-dimensions are the two dual polyhedra pseudoicosahedron-
pyritohedron and the pseudo icosidodecahedron. In quaternionic representations
it finds a natural extension to the 4-dimensions.The related polytopes turn out
to be the pseudo snub 24-cell and its dual polytope expressed in terms of a
parameter x leading to snub 24-cell and its dual in the limit where the
parameter x takes the golden ratio. It turns out that the relevant lattice is
the root lattice of W(D4).
