  Local condition that imply the no-hair property of black holes are completed.
The conditions take the form of constraints on the geometry of the
2-dimensional crossover surface of black hole horizon. They imply also the
axial symmetry without the rigidity theorem. This is the new result contained
in this letter. The family of the solutions to our constraints is 2-dimensional
and can be parametrized by the area and angular momentum. The constraints are
induced by our assumption that the horizon is of the Petrov type D. Our result
applies to all the bifurcated Killing horizons: inner/outer black hole horizons
as well as cosmological horizons. Vacuum spacetimes with a given cosmological
constant can be reconstructed from our solutions via Racz's black hole
holograph.
