This question is regarding the proof of a lemma in the book Reflection groups and Coxeter groups by Humphreys section 6.8.
Lemma: let $E$ be an n-dimensional real vector space endowed with a symmetric bilinear form $B$ of signature $(n-1,1)$. Fix a nonzero vector $\lambda \in E$, and set $H=\{ \mu \in E|B(\lambda, \mu)=0\}$. Then the restriction of $B$ to $H$ is of positive type if and only if $B(\lambda, \lambda) \leq 0$.
Proof. If $B(\lambda, \lambda) \neq 0$, $\lambda$ does not belong to $H$ and so we can write $E=H \oplus \mathbb{R} \lambda$. Now to prove that $B$ restricted to H is non degenerate and is of positive type precisely when $B(\lambda, \lambda) < 0$ because of the hypothesis on the signature of $B$. I was able to prove that if $B$ restricted to $H$ is of positive type then $B(\lambda, \lambda) < 0$. But the otherway Iam not getting the way to prove. please help me.
The form in the basis $(e_1,...,e_n)$ is defined by $B(x,x)=x_1^2+..+x_{n-1}^2-x_n^2$ where $x=x_1e_1+..+x_ne_n$,
Write $\lambda=a_1e_1+..+a_ne_n$, you have $a_1^2+..+a_{n-1}^2<a_n^2$.
Let $x\in H$, $a_1x_1+..+a_{n-1}x_{n-1}=a_nx_n$ write Cauchy-Swartz to $(a_1,..,a_{n-1})$ and $(x_1,..,x_{n-1})$, you have $a^2_nx^2_n=\mid a_1x_1+..a_{n-1}x_{n-1}\mid^2\leq (a_1^2+..a_{n-1}^2)(x_1^2+..+x_{n-1}^2)<a_{n}^2(x_1+..x_{n-1}^2)$. This implies that $x_1+..+x_{n-1}^2>x_n^2$. done.