Let $g$ be a Lie algebra. The classical Yang-Baxter equation (CYBE) is: $$ [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0. $$ The modified classical Yang-Baxter equation (MCYBE) is: $$ [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = \omega, $$ where $\omega \in g \otimes g \otimes g$ is $g$-invariant under the adjoint action.
What is the relation between solutions of classical Yang-Baxter equations and solutions of modified Yang-Baxter equations. If we have a solution $r$ to CYBE, can we obtain a solution to MCYBE? Thank you very much.