  Recently, a new choice of variables was identified to understand how the
quantum group structure appeared in three-dimensional gravity [1]. These
variables are introduced via a canonical transformation generated by a boundary
term. We show that this boundary term can actually be taken to be the volume of
the boundary and that the new variables can be defined in any dimension greater
than three. In addition, we study the associated metric and teleparallel
formalisms. The former is a variant of the Henneaux--Teitelboim model for
unimodular gravity. The latter provides a non-abelian generalization of the
usual abelian teleparallel formulation.
