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Local condition that imply the no-hair property of black holes are completed.The conditions take the form of constraints on the geometry of the2-dimensional crossover surface of black hole horizon. They imply also theaxial symmetry without the rigidity theorem. This is the new result containedin this letter. The family of the solutions to our constraints is 2-dimensionaland can be parametrized by the area and angular momentum. The constraints areinduced by our assumption that the horizon is of the Petrov type D. Our resultapplies to all the bifurcated Killing horizons: inner/outer black hole horizonsas well as cosmological horizons. Vacuum spacetimes with a given cosmologicalconstant can be reconstructed from our solutions via Racz's black holeholograph.

The axial symmetry of Kerr without the rigidity theorem