Let $g$ be a finite dimensional real Lie algebra and $(,)$ be a nondegenerate invariant symmetric bilinear form on $g$. Let $r\in g\bigotimes g$ be a skew-symmetric solution of the MCYBE. We may regard $r$ as a linear map $g^{\star}\mapsto g$, and hence as a linear map $\rho: g\rightarrow g$,identifying $g^{\star}$ with $g$ by using the inner product. $\mathbf{Proposition}$: With the above notation, let $r\in g\otimes g$ be a skew symmetric solution of the MCYBE and define, for $X,Y\in g$, $[X,Y]_{r}=[\rho(X),Y]+[X,\rho(Y)]$. Then $[,]_{r}$ is a Lie bracket on $g$.

I have checked it, but I have problems in proving the Jacobi identity. Can anyone tell me how to prove the Jacobi identity or if there are any useful properties of the linear map $\rho$? Thank you.