  The nonholonomic dynamics can be described by the so-called nonholonomic
bracket in the constrained submanifold, which is a non-integrable modification
of the Poisson bracket of the ambient space, in this case, of the canonical
bracket in the cotangent bundle of the configuration manifold. This bracket was
defined by Cantrijn et al. and Ibort et al., although there was already some
particular and less direct definition. On the other hand, another bracket, also
called noholonomic, was defined using the description of the problem in terms
of almost Lie algebroids. Recently, reviewing two older papers by R. J. Eden,
we have defined a new bracket which we call Eden bracket. In the present paper,
we prove that these three brackets coincide. Moreover, the description of the
nonholonomic bracket \`a la Eden has allowed us to make important advances in
the study of Hamilton-Jacobi theory and the quantization of nonholonomic
systems.
