  We treat a non-normal Fefferman-type construction based on an inclusion
$\SL(n+1)\embed\Spin(n+1,n+1)$. The construction associates a split signature
$(n,n)$-conformal spin structure to a projective structure of dimension $n$.
For $n\geq 3$ 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
