I am reading the book Introduction to lie algebras and representation theory. I have some difficulty in understanding some parts of the book for Freudenthal's formula.
Page 120, line 6, why $m(\mu-i\alpha)=n_0+\cdots+n_i$? Here $m=\langle\mu,\alpha\rangle$ is an integer and $\mu,\alpha$ are weights. So $m(\mu-i\alpha)$ is an element in $H^{*}$ where $H$ is the Cartan subalgebra. But $n_0+\cdots+n_i$ is an integer. How to prove this formula? What does figure 1 on this page mean?
Page 121, section 22.3, line 7, why $(\mu,\mu)=\sum_{i,j}a_ia_j\kappa(h_j,h_i)$? Here $\mu(h_i)=\sum_{j}a_j\kappa(h_j,h_i)$.
Page 123, example 22.4, I cannot compute some of the data in table 1. For example, $(\mu+\delta, \mu+\delta)$. When $\mu=\lambda=\lambda_1+3\lambda_2$, $\delta=(\alpha_1+\alpha_2)/2$, $(\mu+\delta, \mu+\delta)=(13\alpha_1/6+17\alpha_2/6, 13\alpha_1/6+17\alpha_2/6)$. But $(\alpha_1,\alpha_1)=(\alpha_2,\alpha_2)=2$, $(\alpha_1,\alpha_2)=\sqrt{2}\times \sqrt{2}\cos (2\pi/3) = -1$. I cannot get $(\mu+\delta, \mu+\delta)=28/3$.
Thank you very much.