Let $\mathfrak{g}$ be a Lie algebra, $r = \sum_i x_i \otimes y_i \in \mathfrak{g} \otimes \mathfrak{g}.$ Define $r_{12} = \sum_i x_i \otimes y_i \otimes 1,$ $r_{13} = \sum_i x_i \otimes 1 \otimes y_i,$ $r_{23} = \sum_i 1 \otimes x_i \otimes y_i.$ The left part of the Classical Yang-Baxter Equation is $CYB(r) = [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}].$
I am trying to prove that if for a given $r$ $CYB(r)$ = 0, and the symmetric part of $r$ is $ad$ $\mathfrak{g}$ - invariant, then $CYB(\text{skew-symmetric part of } r)$ is $ad$ $\mathfrak{g}$ - invariant.
My idea was to prove that for $r = s + a$, where $s$ is the symmetric part and $a$ the skew-symmetric part of $r$, $CYB(r) = CYB(a) + CYB(s)$. Then if invariance of $s$ implies invariance of $CYB(s)$, we are done. Are these two statements correct? If so, how I can prove them? If there are not, how to solve the original problem?