  Fayos and Sopuerta have recently set up a formalism for studying vacuum
spacetimes with an isometry, a formalism that is centred around the bivector
corresponding to the Killing vector and that adapts the tetrad to the bivector.
Steele has generalized their approach to include the homothetic case. Here, we
generalize this formalism to arbitrary spacetimes and to homothetic and
conformal Killing vectors but do not insist on aligning the tetrad with the
bivector. The most efficient way to use the formalism to find conformal Killing
vectors (proper or not) of a given spacetime is to combine it with the notion
of a preferred tetrad. A metric by Kimura is used as an illustrative example.
