Recent work has demonstrated an attractor mechanism for extremal rotating
black holes subject to the assumption of a near-horizon SO(2,1) symmetry. We
prove the existence of this symmetry for any extremal black hole with the same
number of rotational symmetries as known four and five dimensional solutions
(including black rings). The result is valid for a general two-derivative
theory of gravity coupled to abelian vectors and uncharged scalars, allowing
for a non-trivial scalar potential. We prove that it remains valid in the
presence of higher-derivative corrections. We show that SO(2,1)-symmetric
near-horizon solutions can be analytically continued to give SU(2)-symmetric
black hole solutions. For example, the near-horizon limit of an extremal 5D
Myers-Perry black hole is related by analytic continuation to a non-extremal
cohomogeneity-1 Myers-Perry solution.