A non-normal Fefferman-type construction of split-signature conformal structures admitting twistor spinors

We treat a non-normal Fefferman-type construction based on an inclusion $\text{SL}(n+1)\text{embed}\text{Spin}(n+1,n+1)$. The construction associates a split signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. For $n\ge 3$ the induced conformal Cartan connection is shown to be normal if and only if it is flat. The main technical work of this article consists in showing that in the non-flat case the normalised conformal Cartan connection still allows a parallel (pure) spin-tractor and thus a corresponding (pure) twistor spinor on the conformal space. The Fefferman-type construction presented here is an alternative approach to study a construction of Dunajski-Tod