  We review the classical formulation of general relativity as an SL(2,C) gauge
theory in terms of Ashtekar's selfdual variables and reality conditions for the
spatial metric (RCI) and its evolution (RCII), and we add some new observations
and results. We first explain in detail how a connection taking values in the
Lie algebra of the complex Lorentz group yields two Spin(3,1) connections, one
selfdual and one anti-selfdual, without the need for a spin structure. We then
demonstrate that the selfdual part of the complexified Palatini action in
Ashtekar variables requires a holomorphic phase space description in order to
obtain a non-degenerate symplectic structure. The holomorphic phase space does
not allow for the implementation of the reality conditions as additional
constraints, so they have to be taken care of "by hand" during the
quantisation. We also observe that, due to the action being complex, there is
an overall complex phase that can be chosen at will. We then review the
canonical formulation and the consequences of the implementation of the reality
conditions. We pay close attention to the transformation behaviour of the
various fields under (complex) basis changes, as well as to complex analytic
properties of the relevant functions on phase space.
