I tried to understand Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations. 

In the book [a guide to quantum groups](http://www.amazon.com/Guide-Quantum-Groups-Vyjayanthi-Chari/dp/0521558840), on page 83, there is an example of solutions of the classical Yang-Baxter equation in the case of $\mathfrak{g} = \mathfrak{sl}_3$. 

[![][1]][1]  


  [1]: https://i.sstatic.net/h3Kx1.png

My questions are 

(1) how to compute $t_0$ and $r^0$?

(2) In the case of (b), suppose that
\begin{align}
r^0 = \frac{1}{3} H_{\alpha} \otimes H_{\alpha} + \frac{1}{3} H_{\beta} \otimes H_{\alpha} + \frac{1}{3} H_{\beta} \otimes H_{\beta}.
\end{align} 
I tried to verify that
\begin{align}
r_{12}^0 + r_{21}^0 = t_0, \\
(\alpha \otimes 1)(r^0) + (1 \otimes \beta)(r^0) = 0.
\end{align}
We have 
\begin{align}
r_{12}^0 = r^0 = \frac{1}{3} H_{\alpha} \otimes H_{\alpha} + \frac{1}{3} H_{\beta} \otimes H_{\alpha} + \frac{1}{3} H_{\beta} \otimes H_{\beta}.
\end{align}
I think that
\begin{align}
r_{21}^0 = \tau_{12} r_{12}^0 \tau_{12}.
\end{align}
How to express $r_{21}^0$ using $H_{\alpha}$, $H_{\beta}$?

We have 
\begin{align}
& (\alpha \otimes 1)(r^0)   \\
& = (\alpha \otimes 1)(\frac{1}{3} H_{\alpha} \otimes H_{\alpha} + \frac{1}{3} H_{\beta} \otimes H_{\alpha} + \frac{1}{3} H_{\beta} \otimes H_{\beta}) \\
& = \frac{1}{3} \alpha(H_{\alpha}) \otimes H_{\alpha} + \frac{1}{3} \alpha(H_{\beta}) \otimes H_{\alpha} + \frac{1}{3} \alpha(H_{\beta}) \otimes H_{\beta}.
\end{align}
I think that $\alpha(H_{\alpha})=1$ and $\alpha(H_{\beta})=0$ (is this correct?). Then we have 
\begin{align}
& (\alpha \otimes 1)(r^0)   \\
& = (\alpha \otimes 1)(\frac{1}{3} H_{\alpha} \otimes H_{\alpha} + \frac{1}{3} H_{\beta} \otimes H_{\alpha} + \frac{1}{3} H_{\beta} \otimes H_{\beta}) \\
& = \frac{1}{3} \alpha(H_{\alpha}) \otimes H_{\alpha} + \frac{1}{3} \alpha(H_{\beta}) \otimes H_{\alpha} + \frac{1}{3} \alpha(H_{\beta}) \otimes H_{\beta} \\
& = .\frac{1}{3} \otimes H_{\alpha}. 
\end{align}
Similarly,
\begin{align}
& (1 \otimes \beta)(r^0)   \\
& = .\frac{1}{3} H_{\beta} \otimes 1. 
\end{align}
But we do not have 
\begin{align}
(\alpha \otimes 1)(r^0) + (1 \otimes \beta)(r^0) = 0.
\end{align}
I think that I made some mistake. Any help would be greatly appreciated!