  This article is motivated by the authors interest in the geometry of the Mori
dream space $\mathbb{P}^4$ blown up in $8$ general points. In this article we
develop the necessary technique for determining Weyl orbits of linear cycles
for the four-dimensional case, by explicit computations in the Chow ring of the
resolution of the standard Cremona transformation. In particular, we close this
paper with applications to the question of the dimension of the space global
sections of effective divisors having at most $8$ base points.
