  It was recently conjectured that a certain vacuum Kerr-Schild spacetime,
which may be regarded as a self-dual analog of the Kerr metric, is equivalent
to the self-dual Taub-NUT instanton. We confirm this conjecture by applying the
Cartan-Karlhede algorithm to each metric and showing that for suitable choices
of null tetrad, the algorithm leads to the same invariants and linear isotropy
groups for both, establishing their equivalence. While it is well-known that
the Taub-NUT solution and its self-dual version admit a double Kerr-Schild
form, the observation that the self-dual Taub-NUT instanton admits a single
Kerr-Schild form has only been made very recently. The two metrics we compare
may be regarded as either complex metrics with Lorentzian (1,3) signature or
real metrics with Kleinian (2,2) signature; here we take the latter view.
Significant simplifications occur when the null tetrads are chosen to consist
of two pairs of complex conjugate null vectors rather than four real
independent ones. As a bonus, our work provides the first example of applying
the Cartan-Karlhede algorithm using a null tetrad of this type.
