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 $s_{\alpha^{\vee}}(\beta^{\vee}) = s_{\alpha}(\beta)^{\vee}$.

Lie Groups and Lie AlgebrasCh. VI section 1.1, Lemma 2 and equations (5), (6), and (7) near the end of that section (using that the reflections preserve the chosen inner product by design). Method 2: If you know the result in the reduced case then use that the set of non-divisible roots always forms a reduced root system (use Prop. 13(i) in Ch. VI section 1.4 of Bourbaki, after reading section 1.2). Method 3: Lemma 3.2.4 in the bookPseudo-reductive Groups(2nd ed.), proved there for root data. General advice: read Ch. IV--VI of Bourbaki. It is awesome. $\endgroup$