  We study the interplay between double cross sum decompositions of a given Lie
algebra and classical r-matrices for its semidual. For a class of Lie algebras
which can be obtained by a process of generalised complexification we derive an
expression for classical r-matrices of the semidual Lie bialgebra in terms of
the data which determines the decomposition of the original Lie algebra.
Applied to the local isometry Lie algebras arising in three-dimensional
gravity, decomposition and semidualisation yields the main class of non-trivial
r-matrices for the Euclidean and Poincare group in three dimensions. In
addition, the construction links the r-matrices with the Bianchi classification
of three dimensional real Lie algebras.
