Assuming some acquaintance with the author’s monograph ‘Lectures on the geometry of Poisson manifolds’, Birkhäuser, Basel (1994; Zbl 0810.53019), left and right invariant complementary 2-forms \(\omega\) (defined by the Schouten-Nijenhuis condition \(\{\omega, \omega\} = 0\)) of the Poisson-Lie group \((G, w)\) are studied. In particular, a certain new Lie bracket of vector fields associated to them yields a new Lie algebra structure on the Lie algebra \(\mathcal G\) of \(G\), which is dual to a certain canonical Lie algebra \([\cdot, \cdot]^*\) produced by \(w\) on the dual vector space \({\mathcal G}^*\).