Nearly half-flat $\mathrm{S}\mathrm{U}(3)$-structures on $S^3\times S^3$

We study the $\mathrm{S}\mathrm{U}(3)$-structure induced on an oriented hypersurface of a 7-dimensional manifold with a nearly parallel ${\mathrm{G}}_2$-structure. We call such $\mathrm{S}\mathrm{U}(3)$-structures nearly half-flat. We characterise the left invariant nearly half-flat structures on $S^3\times S^3$. This characterisation then help us to systematically analyse nearly parallel ${\mathrm{G}}_2$-structures on an interval times $S^3\times S^3$.