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

In the book a guide to quantum groups, on page 83, there is an example of solutions of the classical Yang-Baxter equation in the case of $\mathfrak{g} = \mathfrak{sl}_3$.

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!