  In this second paper of the series we continue to spell out a new program for
quantum gravity, grounded in the notion of corner symmetry algebra and its
representations. Here we focus on tetrad gravity and its corner symplectic
potential. We start by performing a detailed decomposition of the various
geometrical quantities appearing in BF theory and tetrad gravity. This provides
a new decomposition of the symplectic potential of BF theory and the simplicity
constraints. We then show that the dynamical variables of the tetrad gravity
corner phase space are the internal normal to the spacetime foliation, which is
conjugated to the boost generator, and the corner coframe field. This allows us
to derive several key results. First, we construct the corner Lorentz charges.
In addition to sphere diffeomorphisms, common to all formulations of gravity,
these charges add a local $\mathfrak{sl}(2,\mathbb{C})$ component to the corner
symmetry algebra of tetrad gravity. Second, we also reveal that the corner
metric satisfies a local $\mathfrak{sl}(2,\mathbb{R})$ algebra, whose Casimir
corresponds to the corner area element. Due to the space-like nature of the
corner metric, this Casimir belongs to the unitary discrete series, and its
spectrum is therefore quantized. This result, which reconciles discreteness of
the area spectrum with Lorentz invariance, is proven in the continuum and
without resorting to a bulk connection. Third, we show that the corner phase
space explains why the simplicity constraints become non-commutative on the
corner. This fact requires a reconciliation between the bulk and corner
symplectic structures, already in the classical continuum theory. Understanding
this leads inevitably to the introduction of edge modes.
