  In this note we construct a first example of a closed 3-form of $\tilde
G_2$-type on $S^3\times S^4$. We prove that $S^3\times S^4$ 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.
