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Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.

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Zagier's one-sentence proof of a theorem of Fermat

This paper by Christian Elsholtz seems to be exactly what you're looking for. It motivates the Zagier/Liouville/Heath-Brown proof and uses the method to prove some other similar statements. Here is …
Daniel Litt's user avatar