  We complete the classification of Hopf algebras whose infinitesimal braiding
is a principal Yetter-Drinfeld realization of a braided vector space of Cartan
type $G_2$ over a cosemisimple Hopf algebra. We develop a general formula for a
class of liftings in which the quantum Serre relations hold. We give a detailed
explanation of the procedure for finding the relations, based on the recent
work of Andruskiewitsch, Angiono and Rossi Bertone.
