computed on Murrumesh (almost idle);
First computation is much faster than second with memory consumption 10 times compared to before;
Second comp. takes 1:30 h, mem consumption is almost factored by 1000.
Additionally, below SINGULAR says "//used time: 4.14 sec". This takes actually about 10 min to determine.

First computation:
************************************************************
> ring r = 0,(x(1),x(2),t(1),dx(1),dx(2),dt(1),u(1)),Dp;
> matrix C[7][7] = 0,1,1,1,1,1,1,0,0,1,1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0;
> matrix D[7][7] = 0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
> def S = nc_algebra(C,D);
> setring S;
> basering;
//   characteristic : 0
//   number of vars : 7
//        block   1 : ordering Dp
//                  : names    x(1) x(2) t(1) dx(1) dx(2) dt(1) u(1)
//        block   2 : ordering C
//   noncommutative relations:
//    dx(1)x(1)=x(1)*dx(1)+1
//    dx(2)x(2)=x(2)*dx(2)+1
//    dt(1)t(1)=t(1)*dt(1)+1
> ideal I = x(1)5*u(1)-5*x(1)4*x(2)*u(1)+10*x(1)3*x(2)2*u(1)-10*x(1)2*x(2)3*u(1)+5*x(1)*x(2)4*u(1)-x(2)5*u(1)-x(1)2*u(1)-2*x(1)*x(2)*u(1)-x(2)2*u(1)+t(1)*u(1),-5*x(1)4*dt(1)*u(1)+20*x(1)3*x(2)*dt(1)*u(1)-30*x(1)2*x(2)2*dt(1)*u(1)+20*x(1)*x(2)3*dt(1)*u(1)-5*x(2)4*dt(1)*u(1)+2*x(1)*dt(1)*u(1)+2*x(2)*dt(1)*u(1)+dx(1)*u(1),5*x(1)4*dt(1)*u(1)-20*x(1)3*x(2)*dt(1)*u(1)+30*x(1)2*x(2)2*dt(1)*u(1)-20*x(1)*x(2)3*dt(1)*u(1)+5*x(2)4*dt(1)*u(1)+2*x(1)*dt(1)*u(1)+2*x(2)*dt(1)*u(1)+dx(2)*u(1);
> memory(0);memory(1);memory(2);
170344
659456
659456
> ideal J = eliminate(I,u(1));
> J;
J[1]=0
> memory(0);memory(1);memory(2);
1621176
2242496
2242496
> listvar(all);
// S                    [0]  *ring
//      J                    [0]  ideal, 1 generator(s)
//      I                    [0]  ideal, 3 generator(s)
// r                    [0]  ring
//      D                    [0]  matrix 7 x 7
//      C                    [0]  matrix 7 x 7
> kill S,r;
> listvar(all);
> memory(0);memory(1);memory(2);
1618856
2242496
2242496
>
************************************************************

Second computation:
************************************************************
> timer = 1;
> ring r = 0,(x(1),x(2),t(1),dx(1),dx(2),dt(1),u(1)),Dp;
> matrix C[7][7] = 0,1,1,1,1,1,1,0,0,1,1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0;
> matrix D[7][7] = 0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
> def S = nc_algebra(C,D);
> setring S;
> basering;
//   characteristic : 0
//   number of vars : 7
//        block   1 : ordering Dp
//                  : names    x(1) x(2) t(1) dx(1) dx(2) dt(1) u(1)
//        block   2 : ordering C
//   noncommutative relations:
//    dx(1)x(1)=x(1)*dx(1)+1
//    dx(2)x(2)=x(2)*dx(2)+1
//    dt(1)t(1)=t(1)*dt(1)+1
> ideal I = x(1)5*u(1)-5*x(1)4*x(2)*u(1)+10*x(1)3*x(2)2*u(1)-10*x(1)2*x(2)3*u(1)+5*x(1)*x(2)4*u(1)-x(2)5*u(1)-x(1)2*u(1)-2*x(1)*x(2)*u(1)-x(2)2*u(1)+t(1)*u(1),-5*x(1)4*dt(1)*u(1)+20*x(1)3*x(2)*dt(1)*u(1)-30*x(1)2*x(2)2*dt(1)*u(1)+20*x(1)*x(2)3*dt(1)*u(1)-5*x(2)4*dt(1)*u(1)+2*x(1)*dt(1)*u(1)+2*x(2)*dt(1)*u(1)+dx(1)*u(1),5*x(1)4*dt(1)*u(1)-20*x(1)3*x(2)*dt(1)*u(1)+30*x(1)2*x(2)2*dt(1)*u(1)-20*x(1)*x(2)3*dt(1)*u(1)+5*x(2)4*dt(1)*u(1)+2*x(1)*dt(1)*u(1)+2*x(2)*dt(1)*u(1)+dx(2)*u(1),-x(1)*x(2)*u(1)+x(1)*x(2);
> ideal J = eliminate(I,u(1));
//used time: 4717.84 sec
> memory(0);memory(1);memory(2);
1308915904
//used time: 4.14 sec
1329631232
1329631232
> listvar(all);
// S                    [0]  *ring
//      J                    [0]  ideal, 60 generator(s)
//      I                    [0]  ideal, 4 generator(s)
// r                    [0]  ring
//      D                    [0]  matrix 7 x 7
//      C                    [0]  matrix 7 x 7
> kill J;
> memory(0);memory(1);memory(2);
1307702232
1329631232
1329631232
> kill S,r;
> listvar(all);
> memory(0);memory(1);memory(2);
1307699920
1329631232
1329631232
>
************************************************************