 Isotropy of nondegenerate ternary quadratic forms over finite fields

Over a finite field of odd cardinality $q$, a diagonal nondegenerate ternary quadratic form $A{x}^2+B{y}^2+C{z}^2$ has a zero with $z=1$: the sets $\{A{x}^2\}$ and $\{-C-B{y}^2\}$ both have $(q+1)/2$ elements and must intersect. This gives a nonzero nonsingular zero. Over a local integer ring with these coefficients units, the Hensel lemma lifts it to an isotropic vector in the fraction field.