  In 1984 LeBrun constructed a CR-twistor space over an arbitrary conformal
Riemannian 3-manifold and proved that the CR-structure is formally integrable.
This twistor construction has been generalized by Rossi in 1985 for
$m$-dimensional Riemannian manifolds endowed with a $(m-1)$-fold vector cross
product (VCP). In 2011 Verbitsky generalized LeBrun's construction of
twistor-spaces to $7$-manifolds endowed with a $G_2$-structure. In this paper
we unify and generalize LeBrun's, Rossi's and Verbitsky's construction of a
CR-twistor space to the case where a Riemannian manifold $(M, g)$ has a VCP
structure. We show that the formal integrability of the CR-structure is
expressed in terms of a torsion tensor on the twistor space, which is a
Grassmanian bundle over $(M, g)$. If the VCP structure on $(M,g)$ is generated
by a $G_2$- or $Spin(7)$-structure, then the vertical component of the torsion
tensor vanishes if and only if $(M, g)$ has constant curvature, and the
horizontal component vanishes if and only if $(M,g)$ is a torsion-free $G_2$ or
$Spin(7)$-manifold. Finally we discuss some open problems.
