We can solve quickly this problem using the basis Grobner theory.

Put $X=[x_{i,j}],Y=[y_{i,j}]$.

We consider -over a field of characteristic $2$- the algebraic system in the $128$ unknowns $x_{i,j},y_{i,j}$ constituted by the equations $M=X+Y,X^3=0,Y^2=Y$ -entrywise- and $x_{i,j}^2=x_{i,j},y_{i,j}^2=y_{i,j}$.

With the help of the FGb library of Maple, we obtain (in 36") that there are no solutions.

We can solve quickly this problem using the basis Grobner theory.

Put $X=[x_{i,j}],Y=[y_{i,j}]$.

We consider -over a field of characteristic $2$- the algebraic system in the $128$ unknowns $x_{i,j},y_{i,j}$ constituted by the equations $M=X+Y,X^3=0,Y^2=Y$ -entrywise- and $x_{i,j}^2=x_{i,j},y_{i,j}^2=y_{i,j}$.

With the help of the FGb library of Maple, we obtain (in 36") that there are no solutions.

$\textbf{Answer to the OP.}$ The maple command "with(Groebner)" is not very efficient; prefer the command "with(FGb)". You need to load the patch here (available only on LINUX)

http://www.mathemagix.org/www/mfgb/doc/html/install_fgb.en.html

Otherwise use "with(Groebner) but it may be a long way.

You can also use "Sage" very powerful but not very practical to use. Here is the program I used.

restart:

with(LinearAlgebra):

n := 8:

N := Matrix([[0, 0, 0, 1], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1]]):

M := DiagonalMatrix([N, N]):

X := Matrix(8, symbol = x):

Y := Matrix(8, symbol = y):

C := M-X-Y: E := X^3: P := Y^2-Y:

K := NULL:

for i to n do

for j to n do

K := K, x[i, j], y[i, j] end do end do:

K := [K]:

F := NULL:

for i to n do

for j to n do

F := F, C[i, j], E[i, j], P[i, j], x[i, j]^2-x[i, j], y[i, j]^2-y[i, j] end do end do:

F := [F]:

with(FGb):

t := time():

solu := fgb_gbasis(F, 2, K, [], {"index" = 10^7, "verb" = 3});

nops(solu):

t3 := time()-t: