Every six-dimensional Λ-vacuum type II(D) spacetime with a non-degenerate generic optical matrix and Weyl fall-off is locally isometric to the doubly-spinning Kerr-NUT-(A)dS metrics with NUTs off, hence type D and Kerr-Schild.
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On the uniqueness of the Kerr-(A)dS metric as a type II(D) solution in six dimensions
Every six-dimensional Λ-vacuum type II(D) spacetime with a non-degenerate generic optical matrix and Weyl fall-off is locally isometric to the doubly-spinning Kerr-NUT-(A)dS metrics with NUTs off, hence type D and Kerr-Schild.