Many questions have been asked on this site about the proof of Lemma 9.1 on page 42 of Humphreys's Lie algebra book. The full proof is posted here. I understand the proof, but I don't like it and would prefer a proof by induction. This seems plausible, since many results like this can be proven by induction, and induction on the dimension of $E$ looks promising. Can anyone offer a proof by induction of this result?
Lemma. Let $\Phi$ be a finite set which spans $E$. Suppose all reflections $\sigma_\alpha$($\alpha \in \Phi$) leave $\Phi$ invariant. If $\sigma \in GL(E)$ leaves $\Phi$ invariant, fixes pointwise a hyperplane $P$ of $E$, and sends some nonzero $\alpha \in \Phi$ to its negative, then $\sigma=\sigma_\alpha$ (and $P=P_\alpha$).