  The flow polytope $\mathcal{F}_{\widetilde{G}}$ is the set of nonnegative
unit flows on the graph $\widetilde{G}$. The subdivision algebra of flow
polytopes prescribes a way to dissect a flow polytope
$\mathcal{F}_{\widetilde{G}}$ into simplices. Such a dissection is encoded by
the terms of the so called reduced form of the monomial $\prod_{(i,j)\in
E(G)}x_{ij}$. We prove that we can use the subdivision algebra of flow
polytopes to construct not only dissections, but also regular flag
triangulations of flow polytopes. We prove that reduced forms in the
subdivision algebra are generalizations of $h$-polynomials of the
triangulations of flow polytopes. We deduce several corollaries of the above
results, most notably proving certain cases of a conjecture of Kirillov about
the nonnegativity of reduced forms in the noncommutative quasi-classical
Yang-Baxter algebra.
