  We define a noncommutative algebra of four basic objects within a
differential calculus on quantum groups: functions, 1-forms, Lie derivatives
and inner derivations, as the cross-product algebra associated with
Woronowicz's (differential) algebra of functions and forms. This definition
properly takes into account the Hopf algebra structure of the Woronowicz
calculus. It also provides a direct proof of the Cartan identity.
