  In this article, we continue the analysis started in \cite{CMT23} for the
matrix code of quadratic relationships associated with a Goppa code. We provide
new sparse and low-rank elements in the matrix code and categorize them
according to their shape. Thanks to this description, we prove that the set of
rank 2 matrices in the matrix codes associated with square-free binary Goppa
codes, i.e. those used in Classic McEiece, is much larger than what is
expected, at least in the case where the Goppa polynomial degree is 2. We build
upon the algebraic determinantal modeling introduced in \cite{CMT23} to derive
a structural attack on these instances. Our method can break in just a few
seconds some recent challenges about key-recovery attacks on the McEliece
cryptosystem, consistently reducing their estimated security level. We also
provide a general method, valid for any Goppa polynomial degree, to transform a
generic pair of support and multiplier into a pair of support and Goppa
polynomial.
