  Isolated Horizons have played an important role in gravitational physics,
from characterization of the endpoint of black hole mergers to black hole
entropy. With an eye towards a canonical formulation we consider general
relativity in first order form. We focus on two issues: i) The role of the
internal gauge freedom in consistent formulations of the action principle, and
ii) the role a 3+1 decomposition has in the allowed internal gauge. We clarify
how the requirement of well posed variational principles compatible with
general weakly isolated horizons (WIHs) does lead to a partial gauge fixing in
the first order descriptions used previously in the literature. We consider the
Palatini action together with the Holst extension, with and without boundary
terms at the horizon. We show that, for the complete configuration space --with
no gauge fixing--, the Palatini action is differentiable without additional
surface terms at the WIH boundary, but the more general Holst action is not. A
surface term at the horizon --that renders the action for asymptotically flat
configurations differentiable-- makes the Holst action differentiable, but only
if one restricts configuration space and partially reduces the internal Lorentz
gauge. For the second issue, we show that upon performing a 3+1 decomposition
and imposing the time gauge, there is a further gauge reduction of the
Hamiltonian theory in terms of Ashtekar-Barbero variables to a $U(1)$-gauge
theory on the horizon. We show that even when the Holst action is
differentiable without additional surface terms or any gauge fixing for Type I
spherically symmetric (strongly) isolated horizons --and a preferred
foliation--, this result does not go through for more general isolated or
weakly isolated horizons. Our results represent the first comprehensive study
of these issues and clarify some contradictory statements found in the
literature.
