Given a G 2 -structure defined on a seven-dimensional manifold, there are many possible ways of making it evolve with the aim of making it torsion-free, easing in turn the search for Riemannian manifolds with holonomy equal to the exceptional Lie group G 2 . Among those evolutions, the so-called isometric flow has the distinctive feature of preserving the underlying metric induced by that G 2 -structure. This flow is built upon the divergence of the full torsion tensor of the flowing G 2 -structures in such a way that its critical points are precisely G 2 -structures with divergence-free full torsion tensor. In this article we study two large families of pairwise nonequivalent non-closed G 2 -structures defined on simply connected solvable Lie groups previously studied in [KL21] and compute the divergence of their full torsion tensor, obtaining that it is identically zero in both cases.