  The polytopes $\mathcal{U}_{I,\bar{J}}$ were introduced by Ceballos, Padrol,
and Sarmiento to provide a geometric approach to the study of
$(I,\bar{J})$-Tamari lattices. They observed a connection between certain
$\mathcal{U}_{I,\bar{J}}$ and acyclic root polytopes, and wondered if
M\'esz\'aros' subdivision algebra can be used to subdivide all
$\mathcal{U}_{I,\bar{J}}$. We answer this in the affirmative from two
perspectives, one using flow polytopes and the other using root polytopes. We
show that $\mathcal{U}_{I,\bar{J}}$ is integrally equivalent to a flow polytope
that can be subdivided using the subdivision algebra. Alternatively, we find a
suitable projection of $\mathcal{U}_{I,\bar{J}}$ to an acyclic root polytope
which allows subdivisions of the root polytope to be lifted back to
$\mathcal{U}_{I,\bar{J}}$. As a consequence, this implies that subdivisions of
$\mathcal{U}_{I,\bar{J}}$ can be obtained with the algebraic interpretation of
using reduced forms of monomials in the subdivision algebra. In addition, we
show that the $(I,\bar{J})$-Tamari complex can be obtained as a triangulated
flow polytope.
