Relation between Pfister forms in W(Q)

Question

For $n$ a nonzero integer, is it possible to have a relation $$ n\langle\langle-1,-1\rangle\rangle=\sum_i n_i\langle\langle a_i,b_i\rangle\rangle$$ in $I^2(\mathbb Q)\subset W(\mathbb Q)$ (or maybe $W(\mathbb R)$ if you want), where $n_i\in\mathbb Z$ and the $a_i,b_i$ are positive rational numbers?

Answer 1

No, it is not possible, neither in $W(\mathbb{R})$ nor in $W(\mathbb{Q})$ : if $a,b>0$ then $\langle\langle a,b\rangle\rangle$ is hyperbolic over $\mathbb{R}$ so is $0$ in $W(\mathbb{R})$, whereas $n\langle\langle -1,-1\rangle\rangle$ is nonzero if $n$ is nonzero. Since the equality cannot hold over $\mathbb{R}$, it cannot hold over $\mathbb{Q}$ either.

Generally speaking, deciding equality in $W(\mathbb{Q})$ is supposed to be easy : you just have to check equality in $W(\mathbb{R})$ and all $W(\mathbb{Q}_p)$, by Hasse-Minskowski's theorem. But $W(\mathbb{R})\simeq \mathbb{Z}$ through signature (easy to compute), and $W(\mathbb{Q}_p)$ is also easy to compute with explicit isomorphisms to finite rings (see Lam's book on quadratic forms for instance).