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I was wondering if anyone knew a good reference for the results in this theorem, or can easily supply a proof themselves. I will explain the notation below.

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This theorem is stated, but not proved, in Linear Algebraic Groups by Armand Borel. Here are the notation/definitions used in the book:

$R$ is a subfield of $\mathbb{R}$. A root system is a pair $(V,\Phi)$, where $V$ is a finite dimensional vector space over $R$, and $\Phi$ is a subset of $V$, such that:

1 . $0 \not\in \Phi$, and $\Phi$ spans $V$.

2 . For each $\alpha \in V$, there exists a (necessarily unique) automorphism $r_{\alpha}$ of $V$ which sends $\alpha$ to $-\alpha$, stabilizes $\Phi$, and fixes pointwise a hyperplane in $V$.

3 . For $\alpha, \beta \in V$, $\beta - r_{\alpha}(\beta)$ is an integral multiple of $\alpha$.

A base of $(V,\Phi)$ is a subset $\Delta$ of $\Phi$ which is a basis for $V$, such that every $\alpha \in \Phi$ can be written as an integer-linear combination of the elements of $\Phi$, with either all nonnegative or all nonpositive coefficients. One can then talk about the positive and negative roots $\Phi^+$ and $\Phi^-$ with respect to a base $\Delta$.

If $\lambda$ is an element of the dual of $V$, we say that $\lambda$ is regular if $\lambda(\alpha) \neq 0$ for any $\alpha \in \Phi$. Define the Weyl chamber of a base $\Delta$: $$\textrm{WC}(\Delta) = \{ \lambda \in V^{\ast} : \lambda(\alpha) > 0, \textrm{ for all } \alpha \in \Delta \}$$ which necessarily consists of regular elements. Given a regular $\lambda$, define $$\Phi^+(\lambda) = \{ \alpha \in \Phi : \lambda(\alpha) > 0 \}$$ and $$\Delta(\lambda) = \{ \alpha \in \Phi^+(\lambda) : \alpha \textrm{ is not the sum of two elements in } \Phi^+(\lambda) \}$$

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I hope not to make mistakes, but it seems to me that this Theorem is proved in the chapter 3 (Rooth Systems), section 10.1 of Humphreys's book "Introduction to Lie Algebras and Representation Theory".

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