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.