  We look at the Poisson structure on the total space of the dual bundle to the
Lie algebroid arising from a matched pair of Lie groups. This dual bundle, with
the natural semidirect product group structure, becomes a Poisson-Lie group as
suggested by a recent work of Stachura. Moreover, when we start from matched
pairs given by the Iwasawa decomposition of simple Lie groups, the associated
Lie bialgebra is coboundary.
