Classification of invariant Poisson brackets on the dual of a simple Lie algebra

Let $\mathfrak{g}$ be a simple Lie algebra. The polynomial algebra on $\mathfrak{g}^*$ is $S\mathfrak{g}$. An invariant Poisson bracket is a Poisson bracket on $S\mathfrak{g}$ invariant under the relevant Lie algebra action. Let $[\cdot,\cdot]$ denote the Poisson–Lie bracket, and let $b$ be an invariant element of $S\mathfrak{g}$. Invariant Poisson bracket classification problem. All invariant Poisson brackets on $S\mathfrak{g}$ should have the form

$$b[\cdot,\cdot].$$

This is posed as a problem in the theory of simple Lie algebras. The supplied text does not state whether it has been solved.