  The Barnich--Troessaert bracket is a proposal for a modified Poisson bracket
on the covariant phase space for general relativity. The new bracket allows us
to compute charges, which are otherwise not integrable. Yet there is a catch.
There is a clear prescription for how to evaluate the new bracket for any such
charge, but little is known how to extend the bracket to the entire phase
space. This is a problem, because not every gravitational observable is also a
charge. In this paper, we propose such an extension. The basic idea is to
remove the radiative data from the covariant phase space. This requires
second-class constraints. Given a few basic assumptions, we show that the
resulting Dirac bracket on the constraint surface is nothing but the BT
bracket. A heuristic argument is given to show that the resulting constraint
surface can only contain gravitational edge modes.
