What is the explicit formula for classical r-matrices?

Question

It is said that classical r-matrices are those satisfy the classical Yang-Baxter equation $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0$, where $r \in \mathfrak{g} \otimes \mathfrak{g}$. I learned from a professor that the following $r$ is a solution of the classical YBE. $$ r = \sum_{1\leq i < j \leq n}^{n} E_{ij} \otimes E_{ji} + \frac{1}{2} \sum_{i=1}^{n} H_i \otimes H_i, $$ where $H_i = E_{ii}$, $E_{ij}$ is a matrix with $1$ at $(i,j)$ and $0$ elsewhere. Are there some other solutions of classical YBE? Are there some general expressions of classical r-matrices? Thank you very much.

Answer 1

Are there some other solutions of classical YBE?

Yes, an early work in this direction can be studied here

A. A. Belavin, V. G. Drinfeld, “Solutions of the classical Yang–Baxter equation for simple Lie algebras”, Funkts. Anal. Prilozh., 16:3 (1982), 1–29

Functional Analysis and Its Applications, 1982, 16:3, 159–180