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.

The original problem is if 
$$M=\begin{bmatrix} N & 0 \\ 0 & N \end{bmatrix}, \mbox{ where } N=\begin{bmatrix} 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 1\end{bmatrix}$$
then $M$ cannot be written as a sum $X+Y$ such that $X^3=0$ and $Y^2=Y.$ Here  $M, X, Y$ are matrices over $\mathbb{F}_2$.

Write $X=[x_{i,j}], Y=[y_{i,j}]$.
For each $(i,j)$, $i,j \in \{1, \ldots, 8\}$, $x_{i,j}$ and $y_{i,j}$ satisfies the following equations:
\begin{equation}
    \begin{cases}
      \displaystyle \sum_{k=1}^8\sum_{l=1}^8 x_{i,k}x_{k,l}x_{l,j}=0\\
      \displaystyle \sum_{k=1}^8 y_{i,k}y_{k,j}=y_{i,j}\\ 
      x_{i,j}+y_{i,j}=m_{i,j} \\
      x_{i,j}^2= x_{i,j} \\
      y_{i,j}^2= y_{i,j}
    \end{cases}\,,
\end{equation}

So we want to show that this system has no solution via Groebner basis and implement it using Maple. I have zero background in Maple so I need help with the code. So what I've researched is we start with
```
with(Groebner);
```
and define matrix $M$:
```
M:=[[0,0,0,1,0,0,0,0], [1,0,0,0,0,0,0,0], [0,1,0,0,0,0,0,0], [0,0,1,1,0,0,0,0], [0,0,0,0,0,0,0,1], [0,0,0,0,1,0,0,0], [0,0,0,0,0,1,0,0], [0,0,0,0,0,0,1,1]];
```
Define the following polynomials for all $(i,j)$:
```
f_{i,j}= \sum_{k=1}^8\sum_{l=1}^8 x_{i,k}x_{k,l}x_{l,j};
g_{i,j}= \sum_{k=1}^8 y_{i,k}y_{k,j}-y_{i,j};
h_{i,j}=x_{i,j}+y_{i,j}-M[i,j];
q_{i,j}=x_{i,j}^2-x_{i,j};
s_{i,j}=y_{i,j}^2-y_{i,j};
```
and we let for all $(i,j)$
```
B:=[f_{i,j},g_{i,j},h_{i,j},q_{i,j},s_{i,j}];
```
and call
```
G:=Groebner[Basis](B, plex(x_{i,j},y_{i,j}), characteristic=2);
```
We expect that the output is 
```
G:=[1]
```
Is this correct? Also, how do I typeset the summations in Maple, and how do I typeset the polynomials so that I don't have to manually type for all 64 pairs $(i,j)$? Thank you so much.