  We extend the definition of the "flipped" loop-quantum-gravity vertex to the
case of a finite Immirzi parameter. We cover the Euclidean as well as the
Lorentzian case. We show that the resulting dynamics is defined on a Hilbert
space isomorphic to the one of loop quantum gravity, and that the area operator
has the same discrete spectrum as in loop quantum gravity. This includes the
correct dependence on the Immirzi parameter, and, remarkably, holds in the
Lorentzian case as well. The ad hoc flip of the symplectic structure that was
initially required to derive the flipped vertex is not anymore needed for
finite Immirzi parameter. These results establish a bridge between canonical
loop quantum gravity and the spinfoam formalism in four dimensions.
