On higher dimensional Einstein spacetimes with a non-degenerate double Weyl aligned null direction

We prove that higher dimensional Einstein spacetimes which possess a geodesic, non-degenerate double Weyl aligned null direction (WAND) $\ell$ must additionally possess a second double WAND (thus being of type D) if either: (a) the Weyl tensor obeys $C_{abc[d}{\ell }_{e]}{\ell }^c=0$ ($\iff {\Phi }_{ij}=0$, i.e. the Weyl type is II(abd)); (b) $\ell$ is twistfree. Some comments about an extension of the Goldberg–Sachs theorem to six dimensions are also made. (paper)