  We examine the self-consistency of the first-order formulation of the
Yang-Mills theory. By comparing the generating functional $Z$ before and after
integrating out the additional field $F^a_{\mu\nu}$, we derive a set of
structural identities that must be satisfied by the Green's functions at all
orders. These identities, which hold in any dimension, are distinct from the
usual Ward identities and are necessary for the internal consistency of the
first-order formalism. They relate the Green's functions involving the fields
$F^a_{\mu\nu}$, to Green's functions in the second-order formulation which
contain the gluon strength tensor $f^a_{\mu\nu}$. In particular, such
identities may provide a simple physical interpretation of the additional field
$F^a_{\mu\nu}$.
