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