  The goal of this note is to first prove that for a well behaved
$\mathbb{Z}^2$-algebra $R$, the category $QGr(R) := Gr(R)/Tors(R)$ is
equivalent to $QGr(R_\Delta)$ where $R_\Delta$ is a diagonal-like
sub-$\mathbb{Z}$-algebra of $R$. Afterwards we use this result to prove that
the $\mathbb{Z}^2$-algebras as introduced in [ArXiV:1607.08383] are
QGr-equivalent to a diagonal-like sub-$\mathbb{Z}$-algebra which is a
simultaneous noncommutative blow-up of a quadratic and a cubic Sklyanin
algebra. As such we link the noncommutative birational transformation and the
associated $\mathbb{Z}^2$-algebras as appearing in the work of Van den Bergh
and Presotto with the noncommutative blowups appearing in the work of Rogalski,
Sierra and Stafford.
