Question regarding isometric relation on Pfister forms

Question

I was going through this note on quadratic forms. While proving lemma 2.16, it is said that since $\pi_n$(a Pfister form) is hyperbolic and $\pi_n=\pi_{n-1}-\langle a_n\rangle\otimes\pi_{n-1}$, $\pi_{n-1}\simeq \langle a_n\rangle\otimes\pi_{n-1}$. How does this follow? Is this because it is being considered in the Witt ring $WF$? In that case will theorem 2.12 in the same note also hold true?

Answer 1

You can show this using just Witt's cancellation theorem. We know $\pi_n = \pi_{n-1} \perp (-a_n)\pi_{n-1}$. Adding to both sides, you get $\pi_n \perp a_n\pi_{n-1} = \pi_{n-1} \perp (-a_n)\pi_{n-1} \perp a_n\pi_{n-1}$. Note since $\pi_n$ is hyperbolic, $\pi_n \simeq 2^{n-1}\mathbb{H}$. Also clearly $(-a_n)\pi_{n-1} \perp a_n\pi_{n-1} \simeq 2^{n-1}\mathbb{H}$. Now by Witt cancellation you obtain $a_n \pi_{n-1} \simeq \pi_{n-1}$.