In Bill Casselman's notes on root systems (<http://www.math.ubc.ca/~cass/courses/tata-07a/Roots.pdf>), I am confused about the proof of the result $s\_{\alpha}(\beta)^{\vee} = s\_{\alpha^{\vee}}(\beta^{\vee})$ for any roots $\alpha,\beta$ (Corollary 3.10). 


In these notes a root system is a quadruple $(V,\Sigma, V^{\vee}, \Sigma^{\vee})$, where $V, V^{\vee}$ are finite dimensional real vector spaces in a perfect pairing, and $\Sigma \subseteq V - 0, \Sigma^{\vee} \subseteq V^{\vee} - 0$ are finite subsets in bijection $\beta \mapsto \beta^{\vee}$. The usual axioms for the root system are given, except we do not require that $\Sigma$ spans $V$.


It is claimed that the result $s\_{\alpha}(\beta)^{\vee} = s\_{\alpha^{\vee}}(\beta^{\vee})$ can be proved in two ways: from Corollary 3.9, and from the orthogonal reflection formula (Corollary 3.3).


I can't see how this follows from Corollary 3.3, nor why Corollary 3.9 is true. 


**Method 1:** Using Corollary 3.9, which states that the coroot $\alpha^{\vee}$ of a root $\alpha$ is the unique element of $V^{\vee}$ which lies in the span of $\Sigma^{\vee}$, satisfies $\langle \alpha, \alpha^{\vee} \rangle = 2$, and satisfies $$\sum\limits\_{\gamma \in (\beta + \mathbb{Z}\alpha) \cap \Sigma} \langle \gamma, \alpha^{\vee} \rangle = 0 $$


for all roots $\beta$. It is claimed that if $l$ is another element of $V^{\vee}$ satisfying these properties, then $l - \alpha^{\vee}$ must be zero, but I don't see why this is.


**Method 2**: For $v, w \in V$, define


$$v \bullet w = \sum\limits\_{\gamma \in \Sigma} \langle v, \gamma^{\vee} \rangle \langle w, \gamma^{\vee} \rangle$$


Then Corollary 3.3 says that for any $v \in V$ and any root $\beta$,


$$\langle v, \beta^{\vee} \rangle = 2 \frac{v \bullet \beta}{\beta \bullet \beta}$$


I don't see why Corollary 3.3 implies that $s\_{\alpha^{\vee}}(\beta^{\vee}) = s\_{\alpha}(\beta)^{\vee}$.



In a non-reduced root system, $s_{\alpha}(\beta)^{\vee} = s_{\alpha^{\vee}}(\beta^{\vee})$
Proof of uniqueness in Method 1: it suffices to show that $\langle \beta, l - \alpha^{\vee} \rangle = 0$ for all roots $\beta$. Write


$$(\beta + \alpha\mathbb{Z}) \cap \Sigma = \{ \beta + n\_i \alpha, i = 1, ... , r \}$$


this is nonempty.


Then 


$$0 = \sum\limits\_{i=1}^r \langle \beta + n\_i \alpha, l \rangle = r \langle \beta, l \rangle + 2(n\_1 + \cdots + n\_r)$$


And also 


$$0 = \sum\limits\_{i=1}^r \langle \beta + n\_i \alpha, \alpha^{\vee}\rangle = r \langle \beta, \alpha^{\vee} \rangle + 2(n\_1 + \cdots + n\_r)$$


so $\langle \beta, l \rangle = \langle \beta , \alpha^{\vee} \rangle$.


