  It is well known that free (unconstrained) initial data for the gravitational
field in general relativity can be identified on an initial hypersurface
consisting of two intersecting null hypersurfaces. Here the phase space of
vacuum general relativity associated with such an initial data hypersurface is
defined; a Poisson bracket is defined, via Peierls' prescription, on
sufficiently regular functions on this phase space, called ``observables''; and
a bracket on initial data is defined so that it reproduces the Peierls bracket
between observables when these are expressed in terms of the initial data. The
brackets between all elements of a free initial data set are calculated
explicitly. The bracket on initial data presented here has all the
characteristics of a Poisson bracket except that it does not satisfy the Jacobi
relations (even though the brackets between the observables do).
  The initial data set used is closely related to that of Sachs. However, one
significant difference is that it includes a ``new'' pair of degrees of freedom
on the intersection of the two null hypersurfaces which are present but quite
hidden in Sachs' formalism. As a step in the calculation an explicit expression
for the symplectic 2-form in terms of these free initial data is obtained.
