Let $r \in \bigwedge^2 \mathfrak {g}$ be a skew-symmetric solution of the CYBE (classical Yang-Baxter equation) so that it gives rise to a triangular structure on $\mathfrak {g}$ i.e. $r$ determines a coboundary structure on $\mathfrak {g}$ with $CYB\ (r) = 0.$ Then the bivector field $\Pi_r^{L}$ defined on the associated simply connected Lie group $G$ defined by $x \mapsto (d_e \lambda_x)^{\otimes 2} (r)$ is a left-invariant non-degenerate Poisson structure on $G$ (where $\lambda_x$ is the left translation by $x \in G$).

I have proved that it is a left-invariant Poisson structure on $G$ but couldn't able to show that the Poisson structure is non-degenerate. Could anyone please help me in this regard?

Thanks for your time.

Source $:$ Lectures on Quantum Groups by Pavel Etingof and Oliver Schiffmann.