The proof of simple roots generating the root systems in T.A. Springer's Linear Algebraic Groups 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.
 A: 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.
