From a Lie algebra g satisfying Z(g) = 0 and Λ 2 (g) g = 0 (in particular, for g semisimple) we describe explicitly all Lie bialgebra structures on extensions of the form L = g × K in terms of Lie bialgebra structures on g (not necessarily factorizable nor quasi-triangular) and its biderivations, for any field K with char K = 0. If moreover, [g, g] = g, then we describe also all Lie bialgebra structures on extensions L = g × K n . In interesting cases we characterize the Lie algebra of biderivations.

We generalize a result on the Heisenberg Lie algebra that gives restrictions to possible Lie bialgebra cobrackets on 2-step nilpotent algebras with some additional properties. For the class of 2-step nilpotent Lie algebras coming from graphs, we describe these extra properties in a very easy graph-combinatorial way.We exhibit applications for f n , the free 2-step nilpotent Lie algebra.