  The Simon tensor gives rise to a local characterization of the Kerr-NUT
family in the stationary class of vacuum spacetimes. We find that a symmetric
and traceless tensor in the quotient space of the stationary Killing trajectory
offers a useful alternative to the Simon tensor. Our tensor is distinct from
the spatial dual of the Simon tensor and illustrates the geometric property of
the three dimensional quotient space more manifest. The reconstruction
procedure of the metric for which the generalized Simon tensor vanishes is
spelled out in detail. We give a four dimensional description of this tensor in
terms of the Coulomb part of the imaginary selfdual Weyl tensor, which
corresponds to the generalization of the three-index tensor defined by Mars.
This allows us to establish a new and simple criterion for the Kerr-NUT family:
the gradient of the Ernst potential becomes the non-null eigenvector of the
Coulomb part of the imaginary selfdual Weyl tensor. We also discuss the ${\rm
SU}(1,2)$ covariant extension of the obstruction tensor into the
Einstein-Maxwell system as an intrinsic characterization of the Kerr-Newman-NUT
family.
