  We give a brief presentation of gwistor space, which is a new concept from
G_2 geometry. Then we compute the characteristic torsion T^c of the gwistor
space of an oriented Riemannian 4-manifold with constant sectional curvature k
and deduce the condition under which T^c is \nabla^c-parallel; this allows for
the classification of the G_2 structure with torsion and the characteristic
holonomy according to known references. The case with the Einstein base
manifold is envisaged.
