Hi, I would like to study a special case of Lie bialgebras. Let $(\mathcal{G},<,>)$ a Lie algebra endowed with a scalar product $<,>$ such that $$\mathcal{G}=S\oplus D(\mathcal{G}),$$ where $S=\lbrace x\in\mathcal{G},\ \mathrm{ad}_x+\mathrm{ad}_x^t=0\rbrace$ is an abelian sub-algebra and $D(\mathcal{G})$, the derived ideal, is abelian of even dimension. (Note that $S$ is the orthogonal of $D(\mathcal{G})$ with respect to $<,>$).

Let $\xi : \mathcal{G}\rightarrow\wedge^2\mathcal{G}$ be a $1$-cocycle with respect to the adjoint representation:

$$\xi([x,y])=\mathrm{ad}_x\xi(y)-\mathrm{ad}_y\xi(x),\ \forall x,y\in\mathcal{G}$$ such that: $$\mathrm{ad}_x\mathrm{ad}_y\xi(z)=0,\ \forall x,y,z\in\mathcal{G}$$ May question is:

How to to characterize the dual bialgebra $\mathcal{G}^*$? It must be of particular type: solvable, unimodular...?

I wonder if such problem was already been studied.

Thanks for your help.