  Two approaches to the tangent space of a noncommutative space whose
coordinate algebra is the enveloping algebra of a Lie algebra are known: the
Heisenberg double construction and the approach via deformed derivatives,
usually defined by procedures involving orderings among noncommutative
coordinates or equivalently involving realizations via formal differential
operators. In an earlier work, we rephrased the deformed derivative approach
introducing certain smash product algebra twisting a semicompleted Weyl
algebra. We show here that the Heisenberg double in the Lie algebra case, is
isomorphic to that product in a nontrivial way, involving a datum $\phi$
parametrizing the orderings or realizations in other approaches. This way, we
show that the two different formalisms, used by different communities, for
introducing the noncommutative phase space for the Lie algebra type
noncommutative spaces are mathematically equivalent.
