  We present a numerical work aiming at the computation of excised initial data
for black hole spacetimes in full general relativity, using Dirac gauge in the
context of a constrained formalism for the Einstein equations. Introducing the
isolated horizon formalism for black hole excision, we especially solve the
non-conformally flat part of the equations, and assess the boundary condition
problem for this part. In the stationary single black hole case, we present and
justify a no-boundary treatment on the black hole horizon. We compare the data
obtained with the well-known analytic Kerr solution in Kerr-Schild coordinates,
and assess the widely used conformally flat approximation for simulating
axisymmetric black hole spacetimes. Our method shows good concordance on
physical and geometrical issues, with the particular application of the
isolated horizon multipolar analysis to confirm that the solution obtained is
indeed the Kerr spacetime. Finally, we discuss a previous suggestion in the
literature for the boundary conditions for the conformal geometry on the
horizon.

General Relativity and Quantum Cosmology