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For a proof, see the appendix.
The reader is warned that often the opposite convention regarding \(\alpha\)- and \(\beta\)-planes is used. The convention used here is chosen to be consistent with [15].
As in the case of \((4,0)\)-signature, a split-signature metric is called self-dual if its Weyl curvature tensor, considered as a bundle-valued 2-form, is its own Hodge-star.