Consider the quaternion algebras $(\frac{-1,-1}{\mathbb{Q}})$ and $(\frac{-1,-3}{\mathbb{Q}})$; it can be shown that these are not isomorphic algebras in following way: we take their tensor product, it is isomorphic to $(\frac{-1,3}{\mathbb{Q}})$ and this product is non-split i.e. it is not isomorphic to matrix ring over $\mathbb{Q}$. Hence we have two elements of Brauer group over $\mathbb{Q}$ whose product is not identity, and so they are non-isomorphic algebras.
Q. Is there any elementary way, not using Brauer's theory or tensor products, to prove non-isomorphism of these two quaternion algebras? Any hint is sufficient.
One way to do it is to consider the norm forms: to any quaternion algebra $\left( \frac{a,b}{\mathbb{Q}} \right)$ we can associate the quadratic form $\langle -a,-b,ab\rangle$, which can be seen as the restriction of the norm to the subspace of pure quaternions.
Now it remains to see that $\langle 1,1,1\rangle$ and $\langle 1,3,3\rangle$ are not isometric over $\mathbb{Q}$. Can you do that ?