ToVector := proc(M) local i,L;
	L := map(op,convert(M,listlist));
	[seq(coeff(i,sqrt(2),0),i=L),seq(coeff(i,sqrt(2),1),i=L)]
end:

ToMatrix := proc(V) local i,j;
	Matrix([seq([seq(V[i+2*j] + V[i+2*j+4]*sqrt(2),i=1..2)],j=0..1)])
end:

w := 2^(1/2);
A1 := Matrix( [[-w , w/2+1],[-2-w,2+w]] );
A2 := Matrix( [[ 1 ,     0],[-2  , 1 ]] );
A3 := Matrix( [[2-w,-w/2+1],[-2+w, w ]] );
A4 := Matrix( [[ -1,  -1  ],[ 0  ,-1 ]] );

G := map(Matrix,[A1, A2, A3, A4,  [[1,0], [0,1]] ] );

B := G:

LengthL := proc(L)
	add( coeff(i,sqrt(2),0)^2 + 2 * coeff(i,sqrt(2),1)^2, i=map(op,L))
end:

to 10 do
	B := {seq(seq( i.j, i=B), j=G)};		# (*)
	lprint(nops(B));
	B := expand(map(convert,B,listlist));
	lprint("RED",nops(B));
	if nops(B) > 10000 then
		B := [seq( [i,LengthL(i)], i = B)]:
		B := sort([op(B)], (i,j) -> i[2] < j[2] )[1..10000];
		B := map( i -> i[1], B)
	fi;
	B := map(Matrix,B):
od:

# Now type B[1..100]  or  B[1..200]  to see the smallest 100 (or 200) matrices in the group.

# Or view some triangular matrices with 1's on the diagonal:

seq(`if`(i[1,1] = 1 and i[2,2] = 1, i, NULL), i=B[1..1000]);