A Quartic Identity Related to Fermat-Type Equations

We provide a short proof of an algebraic identity. For integers $n\ge 2$ and variables $x,y,z$, it represents $(x^n+y^n-z^n)$ as a value of the quadratic form $\mathcal A^2+\mathcal B^2-\mathcal C^2$ after multiplication by an explicit factor. Consequently, any hypothetical solution of $x^n+y^n=z^n$ yields a Pythagorean triple $(\mathcal A,\mathcal B,\mathcal C)$ consisting of explicit polynomial expressions in $x,y,z$.