  Recently, de Thanhoffer de Volcsey and Van den Bergh showed that Grothendieck
groups of "noncommutative Del Pezzo surfaces" with an exceptional sequence of
length 4 are isomorphic to one of three types, the third one not coming from a
commutative Del Pezzo surface. In this paper, we adapt the theory of
noncommutative $\mathbb{P}^1$-bundles as appearing in the work of Van den Bergh
and Nyman to produce a sheaf $\mathbb{Z}$-algebra whose associated Proj has an
exceptional sequence of length 4 for which the Gram matrix is of this third
type. We show that this noncommutative scheme is noetherian and describe its
local structure through the use of our generalized preprojective algebras.
