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In the last few years renewed interest in the 3-tensor potential $L_{abc} $proposed by Lanczos for the Weyl curvature tensor has not only clarified andcorrected Lanczos's original work, but generalised the concept in a number ofways. In this paper we carefully summarise and extend some aspects of theseresults, and clarify some misunderstandings in the literature. We also clarifysome comments in a recent paper by Dolan and Kim; in addition, we correct someinternal inconsistencies in their paper and extend their results.  The following new results are also presented. The (computer checked)complicated second order partial differential equation for the 3-potential, inarbitrary gauge, for Weyl candidates satisfying Bianchi-type equations is given-- in those $n $-dimensional spaces (with arbitrary signature) for which thepotential exists; this is easily specialised to Lanczos potentials for the Weylcurvature tensor. It is found that it is {\it only} in 4-dimensional spaces(with arbitrary signature and gauge), that the non-linear terms disappear andthat the awkward second order derivative terms cancel; for 4-dimensionalspacetimes (with Lorentz signature), this remarkably simple form was originallyfound by Illge, using spinor methods. It is also shown that, for most4-dimensional vacuum spacetimes, any 3-potential in the Lanczos gauges whichsatisfies a simple homogeneous wave equation must be a Lanczos potential forthe Weyl curvature tensor of the background vacuum spacetime. This result isused to prove that the form of a {\it possible} Lanczos potential proposed byDolan and Kim for a class of vacuum spacetimes is in fact a genuine Lanczospotential for these spacetimes.

The Lanczos potential for the Weyl curvature tensor: existence, wave equation and algorithms