Lemma 1

Let $V$ be a vector space over $k$, $R$ a finite subset of $V$ generating $V$. For any $\alpha \in R$ such that $\alpha \neq 0$, there exists at most one reflection $s$ of $V$ such that $s(\alpha) = -\alpha$ and $s(R) = R$.

Let $G$ be the group of automorphisms of $V$ leaving $R$ stable. Since $R$ generates $V$, $G$ is isomorphic to a subgroup of the symmetric group of $R$, and hence is finite. Let $s, s'$ be reflections of $V$ such that $s(\alpha) = s'(\alpha) = -\alpha$, $s(R) = R$, $s'(R) = R$. Then $t = ss'$ belongs to $G$, and hence is of finite order $m$. On the other hand,

Hence, there exists a linear form $f$ on $V$ such that

and $f(\alpha) = 0$. By induction on $n$, it follows that

Taking $n$ equal to $m$, we see that $mf(x) = 0$ for all $x \in V$, so $f = 0$, $t = 1$ and $s = s'$.