A simple construction of potential operators for compensated compactness

We give a short proof of the fact that each homogeneous linear differential operator $\mathscr{A}$ of constant rank admits a homogeneous potential operator $\mathscr{B}$, meaning that $$\ker\mathscr{A}(\xi)=\mathrm{im\,}\mathscr{B}(\xi) \quad\text{for }\xi\in\mathbb{R}^n\backslash\{0\}.$$ We make some refinements of the original result and some related remarks.