  In the context of non-geometric type II orientifold compactifications, there
have been two formulations for representing the various NS-NS
Bianchi-identities. In the first formulation, the standard three-form flux
($H_3$), the geometric flux ($\omega$) and the non-geometric fluxes ($Q$ and
$R$) are expressed by using the real six-dimensional indices (e.g. $H_{ijk},
\omega_{ij}{}^k, Q_i{}^{jk}$ and $R^{ijk}$), and this formulation has been
heavily utilized for simplifying the scalar potentials in
toroidal-orientifolds. On the other hand, relevant for the studies beyond
toroidal backgrounds, a second formulation is utilized in which all flux
components are written in terms of various involutively even/odd $(2,1)$- and
$(1,1)$-cohomologies of the complex threefold. In the lights of recent model
building interests and some observations made in arXiv:0705.3410 and
arXiv:0709.2186, in this article, we revisit two most commonly studied toroidal
examples in detail to illustrate that the present forms of these two
formulations are not completely equivalent. To demonstrate the same, we
translate all the identities of the first formulation into cohomology
ingredients, and after a tedious reshuffling of the subsequent constraints,
interestingly we find that all the identities of the second formulation are
embedded into the first formulation which has some additional constraints. In
addition, we look for the possible solutions of these Bianchi identities in a
detailed analysis, and we find that some solutions can reduce the size of
scalar potential very significantly, and in some cases are too strong to break
the no-scale structure completely. Finally, we also comment on the influence of
imposing some of the solutions of Bianchi identities in studying moduli
stabilization.
