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