In the very recent paper [15], the second author proved that for any {f\in L^{2}(\mathbb{R}^{n},\mathbb{R}^{N})} , the fully nonlinear first order system {F(\,\cdot\"\mathrm{D}u)=f} is well posed in the so-called J. L. Lions space and, moreover, the unique strong solution {u\colon\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}} to the problem satisfies a quantitative estimate. A central ingredient in the proof was the introduction of an appropriate notion of ellipticity for F inspired by Campanato's

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... ssical work in the 2nd order case. Herein, we extend the results of [15] by introducing a new strictly weaker ellipticity condition and by proving well-posedness in the same "energy" space.