  We study a bordism relation for stable 3-forms on a 6-manifold, which is a
binary relation on the set of closed $SL(3;\mathbb{C})$-structures on a
6-manifold via closed $G_2$-structures. Under $SO(3)$-symmetry and a
co-associative condition the relation is reduced to a relation for geometric
structures on a 3-manifold. Under these conditions we prove that the bordism
relation is irreflexive and that the relation induces a more rigid one.
