  A geometric study for an integrable 4-dimensional dynamical system so called
the Fuji-Suzuki-Tsuda system is given. By the resolution of indeterminacy, the
group of its B\"aklund transformations is lifted to a group of
pseudo-isomorphisms between rational varieties obtained from $(P^1)^4$ by
blowing-up along eight 2-dimensional subvarieties and four 1-dimensional
subvarieties. The root basis is realised in the N\'eron-Severi bilattices. A
discrete Painlev\'e system with quadratic degree growth is also realised as its
translational element.
