  An alternative to the representation of complex relativity by self-dual
complex 2-forms on the spacetime manifold is presented by assuming that that
the bundle of real 2-forms is given an almost-complex structure. From this, one
can define a complex orthogonal structure on the bundle of 2-forms, which
results in a more direct representation of the complex orthogonal group in
three complex dimensions. The geometrical foundations of general relativity are
then presented in terms of the bundle of oriented complex orthogonal 3-frames
on the bundle of 2-forms in a manner that essentially parallels their
construction in terms of self-dual complex 2-forms. It is shown that one can
still discuss the Debever-Penrose classification of the Riemannian curvature
tensor in terms of the representation presented here.
