We extend the Fefferman-Phong inequality to N x N systems of PDEs with symbol p(x, xi) = A(x)e(xi) + B(x, xi) + C(x) = p(x, xi)* > or = -cl, (x, xi)[abstract: see text] where e is a positive homogeneous quadratic form, and B(x, xi) = Sigma(l=1)(n) B(l)(x)xi(l). We thus generalize a result by L.-Y. Sung that was obtained for systems of ordinary differential equations. Our proof exploits a Calderón-Zygmund decomposition of the phase-space )[abstract: see text] of the kind introduced by C. Fefferman and D. H. Phong for studying subelliptic differential operators and goes by induction on the size N of the system.