reserve z1,z2,z3,z4,z for Quaternion;

theorem
  <i>^3=-<i>
proof
  <i> = [*0,jj,0,0*] by Lm2,QUATERNI:def 6;then
  <i> * <i> *<i>=( [* 0*0-1*1-0*0-0*0, 0*1+1*0+0*0-0*0, 0*0+0*0+1*0-0*1, 0
  *0+0*0+1*0-0*1 *]) *[*0,1,0,0*] by QUATERNI:def 10
    .=[* (-1)*0-0*1-0*0-0*0,(-1)*1+0*0+0*0-0*0, (-1)*0+0*0+1*0-0*0,(-1)*0+0*
  0+0*0-0*jj *] by QUATERNI:def 10
    .=-[*0,jj,0,0*] by QUATERN2:4;
  hence thesis by Lm2,QUATERNI:def 6;
end;
