In Springer's book Linear Algebraic Groups, the author presented Theorem 8.2.8. about simple roots, root system and Weyl group, as below:
Here $R$ is the root system, $D$ is the set of simple roots, $W$ is the Weyl group.
$R^+$ is the set of positive root (The definition in this book is if there is an element $x$ in the real vector space $V$ generated by $R$ such that $(x,\alpha)\neq 0$ for all $\alpha\in R$, and $R^+$ is the set of $\alpha$ such that $(x,\alpha)>0$.) $D$ is the set of simple roots. ($\alpha\in R^+$ lies in $D$ if and only if $s_\alpha.R^+$ and $R^+$ are adjacent.)
In the proof I think there are two issues:
First, in the third paragraph, the author claimed that if $\beta\in D$ and $\alpha\in R^+-D$, then $s_\beta.\alpha\in R^+-D$. However, one can easy find that this can be wrong in the root system $A_2$: let $\alpha = (1,0)$ and $\beta = (-1/2,\sqrt3/2)$, then $s_\alpha(\beta+\alpha) = \beta$.
Second, in the fourth paragraph, the author wrote $$\alpha = \sum_{\beta\in D}c_\beta \beta+t$$ for some real $c_\beta$ and $t$ lying in the subspace $V_0$ orthogonal to all the simple roots. He said that $W^{'}$ (the group generated by simple reflection, which have not been proved to be the whole group $W$ yet) stabilizes the subspace generated by simple roots and fixed the vector in $V_0$.
So far it is OK. However, he next claimed that apply $s_\alpha$ on both sides of the formula we have $t = 0$. I guess he thought that $s_\alpha$ fixes $t$ and came to the conclusion. However, the proof of (i) depends on this result...
Thanks for any help.
It is clear that the first question is just a typo.
Today I figure out the details behind the second question. If $\alpha = \sum_{\beta\in D}c_\beta\beta+t$, with real coefficients, where $t$ lies in the subspace $V_0$ of $V$ orthogonal to the subspace generated by all $\beta\in D$. Apply $s_\alpha$ to both side we have $$-\alpha = \sum_{\beta\in D}c_\beta\beta-\alpha\sum_{\beta\in D}c_\beta\langle\beta,\alpha^\vee\rangle+s_\alpha t = \alpha-\alpha\sum_{\beta\in D}\langle\beta,\alpha^\vee\rangle+s_\alpha t,$$ and $s_\alpha t = \lambda\alpha$ for some $\lambda\in\mathbb R$. Apply $s_\alpha$ again we have $t = -\lambda\alpha$ ($s_\alpha$ is of order $2$). Note that $\lambda$ can not be $-1$ otherwise we have all $c_\beta=0$ since $\beta\in D$ are linearly independent. Then by substituting $t = -\lambda\alpha$ to the above formula of $\alpha$ we may assume that $\alpha = \sum_{\beta\in D}c_\beta\beta$.
Here I do not show $t = 0$ directly, but from what I write we can assume that $\alpha$ is a linear combination of simple roots with real coefficients.