Question: Let $\mathfrak d = \mathfrak g\bowtie \mathfrak g^*$ be the double of the Lie bialgebra $(\mathfrak g, \mathfrak g^*)$, and let $\mathfrak h$ be a Lie subalgebra of $\mathfrak g$. If $\xi,\eta\in\mathfrak g^*$ such that
$$ \xi(x)=\eta(x)=0, \quad \forall x\in \mathfrak h $$
then is it possible to prove that $([\xi,\eta]_{\mathfrak g^*})(x)=0,\,\forall x\in\mathfrak h$ as well?
This seems to be an important part in proving Theorem 1.1 in V.G. Drinfeld's article "On Poisson homogeneous spaces of Poisson-Lie groups", but I'm having trouble reproducing it. The article itself says "The proof of Theorem 1 is simple and more or less straightforward", so does anyone agree with the author?
Article link: http://www.mathnet.ru/links/e57aaacb0b7a60933c6976d46cd746c2/tmf1461.pdf