I want to prove that in an irreducible root system $\Phi$, there are at most $2$ possible lengths for the roots of $\Phi$.
Just to clarify, I have seen both posts Possible lenghts in an irreducible root system and Two questions on roots of finite, simple, complex lie algebra.
The problem is that my definition of a root system $\Phi$ is of a finite subset of the vector space $\mathbb{R}^{n}$ such that $0\notin\Phi$ and that for all $\alpha\in\Phi$
- for $\lambda\in\mathbb{R}$, it holds that $\lambda\alpha\in\Phi\iff \lambda=\pm 1$; and
- $\Phi$ is invariant under the reflections $\sigma_{\alpha}$ defined by $\alpha$.
For the sake of completeness, $\Phi$ is called irreducible if it cannot be written as $\Phi=\Phi'\sqcup\Phi''$, for some root systems $\Phi',\Phi''$.
Therefore, I don't have the assumption $2\langle α,β\rangle/\langle α,α\rangle\in\mathbb{Z}$ for each $\alpha,\beta\in\Phi$, which is essential(?) in the argument of Humphreys in "Introduction to Lie Algebras and Representation Theory, §10.4, Lemma C". Here $\langle\cdot,\cdot\rangle$ is the usual inner product of $\mathbb{R}^{n}$. In that same book, there is a lemma that states the following:
Let $\Phi$ be an irreducible system, $W=\langle s_{\alpha}\colon \alpha\in\Phi\rangle$ be the Weyl group of $\Phi$, and $E=\mathrm{span}(\Phi)$. Then, if $\alpha\in\Phi$ is a root, the $W$-orbit $W\alpha$ of $\alpha$ spans $E$.
This can be proved using the axioms that I have and also I think it will be useful. For example for proving that for every $\alpha,\beta\in\Phi$ there is $s\in W$ such that $\langle s(\alpha),\beta\rangle\neq 0$.
However, I have no idea how to proceed.
