The existence of closed 3-forms of ${\tilde{G}}_2$-type on 7-manifolds

In this note we construct a first example of a closed 3-form of $\tilde G_2$-typeon$S^3\times S^4$\mathrm{.}Weprovethat$S^3\times S^4$ does not admit a homogeneous 3-form of ${\tilde{G}}_2$-type. Thus our example is a first example of a closed 3-form of ${\tilde{G}}_2$-type on a compact 7-manifold which is not stably homogeneous.