Let $(\mathcal{G}, \mathcal{G}^*, \delta)$ be a Lie bialgebra. Suppose that the structure constant on $\mathcal{G}^*$ and $\mathcal{G}$ are \begin{align} & [t^a, t^b]_* = C_c^{ab} t_c, \\ & [t_a, t_b] = f_{ab}^c t_c, \end{align} respectively.
Let $r = r^{ab} t_a \otimes t_b \in \mathcal{G} \otimes \mathcal{G}$ be a classical r-matrix and assume that $\delta: \mathcal{G} \to \mathcal{G} \wedge \mathcal{G}$ is given by $\delta(X) = [X \otimes 1 + 1 \otimes X, r]$, $X \in \mathcal{G}$.
In the paper, it is said that the structure constant on $\mathcal{G}^*$ is $C_{c}^{ab} = f_{cd}^b r^{ad} + f_{cd}^a r^{db}$.
How to show that the structure constant on $\mathcal{G}^*$ is $C_{c}^{ab} = f_{cd}^b r^{ad} + f_{cd}^a r^{db}$?
Thank you very much.