  We study two properties of nonsingular and infinite measure-preserving
ergodic systems: weak double ergodicity, and ergodicity with isometric
coefficients. We show that there exist infinite measure-preserving
transformations that are ergodic with isometric coefficients but are not weakly
doubly ergodic. We also give type $\text{III}_\lambda$ examples of such
systems, $0<\lambda\leq 1$. We prove that under certain hypotheses, systems
that are weakly mixing are ergodic with isometric coefficients and along the
way we give an example of a uniformly rigid topological dynamical system along
the sequence $(n_i)$ that is not measure theoretically rigid along $(n_i)$ for
any nonsingular ergodic finite measure.
