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.



The Jacobi identity of a Lie algebra?
Write the modified CYBE as $B\_R(x,y)+\lambda [x,y]=0$ with 
$$
B\_R(x,y):=[R(x),R(y)]-R([R(x),y]+[x,R(y)]),
$$
and $\lambda\in \mathbb{R}$. Then
note that the bracket $[x,y]\_R:=[R(x),y]+[x,R(y)]$ satisfies the Jacobi identity if and only if
$$
[B\_R(x,y),w]+[B\_R(y,w),x]+[B\_R(w,x),y]=0
$$
for all $x,y,w\in \mathfrak{g}$.


