  The vertex connecting one background gluon with two quantum ones constitutes
a central ingredient in the gauge-invariant Schwinger-Dyson equation that
determines the non-perturbative dynamics of the gluon propagator. This vertex
satisfies a Ward identity with respect to the background gluon, and a
Slavnov-Taylor identity with respect to the two quantum gluons. We present a
complete Ansatz for this vertex, which satisfies both aforementioned
identities. This entire construction depends crucially on a set of constraints
relating the various form-factors of the ghost Green's functions appearing in
the Slavnov-Taylor identity satisfied by the vertex. The validity of these
constraints is demonstrated to all orders.
