I am working on a problem involving nilpotent matrices over $\mathbb{F}_2$ and I was able to reduce it to proving that the system \begin{equation} \begin{cases} A^2+ BC+ BCA+ ABC+A = I_4 \\ AB+ABD+BCB = 0 \\ CA+DCA+CBC = 0 \\ DCB+CBD = I_4 \\ A^3+BCA+ABC+BDC=0 \\ A^2B+BCB+ABD+BD^2=0 \\ CA^2+DCA+CBC+D^2C=0 \\ CAB+DCB+CBD+D^3=0 \end{cases}\,, \end{equation} has no solution, where $A, B, C, D$ are $4 \times 4$ matrices over $\mathbb{F}_2.$ The first four equations were obtained by plugging-in matrices to some polynomial in $\mathbb{F}_2$ while the other four came from the condition that $$\left[ \begin{array}{c|c} A & B \\ \hline C & D \end{array} \right]^3 = 0.$$

Any advice on how to prove this? Any help would be appreciated.