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

(Conformal) Killing vectors and their associated bivectors