I have a question about the equation (1.24) in [the paper](http://arxiv.org/pdf/math/9910188.pdf) about classical r-matrices.


It is said that when we put $\overline{r} = Pr$ in the equation (1.24):
$$
\overline{r}\_{23}\overline{r}\_{12}P\_{23} + \overline{r}\_{23}P\_{12}\overline{r}\_{23} + P\_{23}\overline{r}\_{12}\overline{r}\_{23} = \overline{r}\_{12}\overline{r}\_{23}P\_{12} + \overline{r}\_{12}P\_{23}\overline{r}\_{12} + P\_{12}\overline{r}\_{23}\overline{r}\_{12},
$$
then we obtain the classical Yang-Baxter equation (1.16)
$$
[r\_{12}, r\_{13}] + [r\_{13}, r\_{23}] + [r\_{12}, r\_{23}] = 0.
$$
I try to prove this statement. But I encounter some problems.


I think that we have 
$$
(Pr)\_{23} = P\_{23} r\_{23} P\_{23} = P\_{32}.
$$


Therefore 
\begin{align}
& \overline{r}\_{23} \overline{r}\_{12} P\_{23} \\
& = P\_{23} r\_{23} P\_{23} P\_{12} r\_{12} P\_{12} P\_{23} \\
& = r\_{32} r\_{21} P\_{23}.
\end{align}


But I don't know how to remove $P$ in $r\_{32} r\_{21} P\_{23}$. How to prove the statement: when we put $\overline{r} = Pr$ in the equation (1.24), then we obtain the classical Yang-Baxter equation (1.16)? Thank you very much.



How to obtain the classical Yang-Baxter equation from a related equation
The index notation refers to the tensor factors these maps act on, i.e. $(Pr)\_{23}=P\_{23}r\_{23}$ etc. To prove the claim, you collect all P factors on the left, where they give the total inversion permutation M on both sides of the equation. Multiplying by M from the left then gives the CYBE.


