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Remark 1: I think of the above fact in terms of the ternary norm form and the quaternary norm form associated to a quaternion algebra over $K$. See e.g. Theorem 95 in these notes on non-commutative algebra for a proof. But in fact if you look at the proof you see that quaternion algebras are not mentioned, so it really is an elementary but not completely trivial fact from the algebraic theory of quadratic forms.

Remark 1: I think of the above fact in terms of the ternary norm form and the quaternary norm form associated to a quaternion algebra over $K$. See e.g. Theorem 95 in these notes on non-commutative algebra for a proof. But in fact if you look at the proof you see that quaternion algebras are not mentioned, so it really is an elementary but not completely trivial fact from the algebraic theory of quadratic forms.

[Remark 4: The argument given by Conway in Will Jagy's answer gives a more elementary solution to Lam's exercise in the above remark than the one I referred to. Namely, if $q$ represents $-\operatorname{disc}(q)$, then $q \cong [-\operatorname{disc}(q),a,b]$, and checking discriminants gives $\operatorname{disc}([a,b]) = -1$. But a binary form has discriminant $-1$ iff it is isomorphic to the hyperbolic plane, so this implies that $q$ is isotropic.]

Finally, my question: what is the "classical proof" of the above theorem about positive-definite ternary integral quadratic forms? How far back does this result go? I would not be at all surprised to learn that it was known to Lagrange / Legendre / Gauss, but presumably they would not have proved it as I did above!

Finally, my question: what is the "classical proof" of the above theorem about positive-definite ternary integral quadratic forms? How far back does this result go? I would not be at all surprised to learn that it was known to Lagrange / Legendre / Gauss, but presumably they would not have proved it as I did above!