  We give a constructive account of the fundamental ingredients of Poisson Lie
theory as the basis for a description of the classical double group $D$. The
double of a group $G$ has a pointwise decomposition $D\sim G\times G^*$, where
$G$ and $G^*$ are Lie subgroups generated by dual Lie algebras which form a Lie
bialgebra. The double is an example of a factorisable Poisson Lie group, in the
sense of Reshetikhin and Semenov-Tian-Shansky [1], and usually the study of its
Poisson structures is developed only in the case when the subgroup $G$ is
itself factorisable. We give an explicit description of the Poisson Lie
structure of the double without invoking this assumption. This is achieved by a
direct calculation, in infinitesimal form, of the dressing actions of the
subgroups on each other, and provides a new and general derivation of the
Poisson Lie structure on the group $G^*$. For the example of the double of
SU(2), the symplectic leaves of the Poisson Lie structures on SU(2) and
SU(2$)^*$ are displayed.
