  The principal aim of this paper is to derive an abstract form of the third
Green identity associated with a proper extension $T$ of a symmetric operator
$S$ in a Hilbert space $\mathfrak H$, employing the technique of quasi boundary
triples for $T$. The general results are illustrated with couplings of
Schr\"{o}dinger operators on Lipschitz domains on smooth, boundaryless
Riemannian manifolds.
