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 $C_{abc[d}ell_{e]}ell^c=0$ ($LeftrightarrowPhi_{ij}=0$, 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.