  It is well-known that a nonsingular minimal cubic surface is birationally
rigid; the group of its birational selfmaps is generated by biregular selfmaps
and birational involutions such that all relations between the latter are
implied by standard relations between reflections on an elliptic curve. It is
also known that a factorial nodal quartic threefold is birationally rigid and
its group of birational selfmaps is generated by biregular ones and certain
birational involutions. We prove that all relations between these involutions
are implied by standard relations on elliptic curves, complete the proof of
birational rigidity over a non-closed field and describe the situations when
some of the birational involutions in question become regular (and, in
particular, complete the proof of the initial theorem on birational rigidity,
since some details were not established in the original paper of M.Mella).
