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

Space of initial conditions for the four-dimensional Fuji-Suzuki-Tsuda system