  We show that, if there exists a realization of a Hopf algebra $H$ in a
$H$-module algebra $A$, then one can split their cross-product into the tensor
product algebra of $A$ itself with a subalgebra isomorphic to $H$ and commuting
with $A$. This result applies in particular to the algebra underlying
inhomogeneous quantum groups like the Euclidean ones, which are obtained as
cross-products of the quantum Euclidean spaces $R_q^N$ with the quantum groups
of rotation $U_qso(N)$ of $R_q^N$, for which it has no classical analog.
