I am reading the book. I have some questions about the computations in (4.1) on page 40.
The computation are in the following.
I don't know why $$ ([[e_r^*, e_k], e_s^*]+[e_r^*, [e_s^*,e_k]],e_l) \\ = \sum_t \alpha_{kt}^r \beta_{l}^{ts} + \sum_t \alpha_{tl}^s \beta_{k}^{rt} + \sum_t \alpha_{kt}^s \beta_{l}^{rt} + \sum_t \alpha_{tl}^r \beta_{k}^{st}. $$
It seems that the formula for $[e_i^*, e_j]$ has not been given.
Thank you very much.
We have $$ ([e_i, e_j^*], e_k^*) = (e_i, [e_j^*, e_k^*]) = (e_i, \sum_s \beta_s^{jk} e_s^*) = \beta_i^{jk}. $$ Therefore $$ [e_i, e_j^*] = \sum_k \beta_i^{jk} e_k + \text{ something in } \mathfrak{g}^*. $$
On the other hand, $$ ([e_i, e_j^*], e_k) = -(e_j^*, [e_i, e_k]) = (e_j^*, \sum_s \alpha_{ik}^{s} e_s) = \alpha_{ik}^{s}. $$ Therefore $$ [e_i, e_j^*] = \sum_k \beta_i^{jk} e_k - \sum_{k} \alpha_{ik}^j e_k^*. $$