{OrePolynomial[{{b^4 - 2*b^2*d^2 + d^4, {2, 0}}, {-3*b^4 + 6*b^2*d^2 - 3*d^4, 
    {1, 1}}, {b^4 - 2*b^2*d^2 + d^4, {0, 2}}, 
   {3*b^3 - 4*b^5 + 2*b^2*d + 3*b^4*d - 3*b*d^2 + 8*b^3*d^2 - 2*d^3 - 
     6*b^2*d^3 - 4*b*d^4 + 3*d^5, {1, 0}}, 
   {-2*b^3 + 3*b^5 - 3*b^2*d - 4*b^4*d + 2*b*d^2 - 6*b^3*d^2 + 3*d^3 + 
     8*b^2*d^3 + 3*b*d^4 - 4*d^5, {0, 1}}, 
   {4*b^2 + 2*b^6 + 12*b*d - 6*b^5*d + 4*d^2 - 2*b^4*d^2 + 12*b^3*d^3 - 
     2*b^2*d^4 - 6*b*d^5 + 2*d^6, {0, 0}}}, 
  OreAlgebraObject[{Der[d], Der[b]}, Expand, #1 + #2 & , Expand[#1*#2] & , 
   None], DegreeLexicographic], OrePolynomial[
  {{-7*b^7 - 3*b^6*d + 21*b^5*d^2 + 9*b^4*d^3 - 21*b^3*d^4 - 9*b^2*d^5 + 
     7*b*d^6 + 3*d^7, {1, 2}}, {3*b^7 + 2*b^6*d - 9*b^5*d^2 - 6*b^4*d^3 + 
     9*b^3*d^4 + 6*b^2*d^5 - 3*b*d^6 - 2*d^7, {0, 3}}, 
   {12*b^6 - 18*b^8 - 8*b^7*d - 26*b^4*d^2 + 54*b^6*d^2 + 24*b^5*d^3 + 
     16*b^2*d^4 - 54*b^4*d^4 - 24*b^3*d^5 - 2*d^6 + 18*b^2*d^6 + 8*b*d^7, {1, 
    1}}, {-9*b^6 + 13*b^8 - 13*b^5*d + 15*b^4*d^2 - 43*b^6*d^2 + 26*b^3*d^3 - 
     3*b^2*d^4 + 51*b^4*d^4 - 13*b*d^5 - 3*d^6 - 25*b^2*d^6 + 4*d^8, {0, 2}}, 
   {-12*b^5 + 12*b^7 - 12*b^9 + 3*b^4*d - 6*b^8*d + 9*b^3*d^2 - 24*b^5*d^2 + 
     36*b^7*d^2 - 6*b^2*d^3 + 18*b^6*d^3 + 3*b*d^4 + 12*b^3*d^4 - 
     36*b^5*d^4 + 3*d^5 - 18*b^4*d^5 + 12*b^3*d^6 + 6*b^2*d^7, {1, 0}}, 
   {21*b^5 - 16*b^7 + 18*b^9 + 61*b^4*d - 18*b^6*d - 12*b^8*d + 6*b^3*d^2 + 
     14*b^5*d^2 - 66*b^7*d^2 - 55*b^2*d^3 + 30*b^4*d^3 + 36*b^6*d^3 - 
     27*b*d^4 + 20*b^3*d^4 + 90*b^5*d^4 - 6*d^5 - 6*b^2*d^5 - 36*b^4*d^5 - 
     18*b*d^6 - 54*b^3*d^6 - 6*d^7 + 12*b^2*d^7 + 12*b*d^8, {0, 1}}, 
   {-24*b^4 + 24*b^6 - 6*b^8 + 8*b^10 - 120*b^3*d + 72*b^5*d - 6*b^7*d - 
     12*b^9*d - 60*b^2*d^2 + 28*b^4*d^2 - 6*b^6*d^2 - 34*b^8*d^2 - 44*b*d^3 - 
     72*b^3*d^3 + 8*b^5*d^3 + 36*b^7*d^3 - 12*d^4 - 60*b^2*d^4 + 30*b^4*d^4 + 
     54*b^6*d^4 + 2*b^3*d^5 - 36*b^5*d^5 + 8*d^6 - 18*b^2*d^6 - 38*b^4*d^6 - 
     4*b*d^7 + 12*b^3*d^7 + 10*b^2*d^8, {0, 0}}}, 
  OreAlgebraObject[{Der[d], Der[b]}, Expand, #1 + #2 & , Expand[#1*#2] & , 
   None], DegreeLexicographic], OrePolynomial[
  {{35*b^9 + 15*b^8*d - 140*b^7*d^2 - 60*b^6*d^3 + 210*b^5*d^4 + 90*b^4*d^5 - 
     140*b^3*d^6 - 60*b^2*d^7 + 35*b*d^8 + 15*d^9, {0, 4}}, 
   {-255*b^8 + 204*b^10 - 120*b^7*d + 72*b^9*d + 790*b^6*d^2 - 822*b^8*d^2 + 
     360*b^5*d^3 - 288*b^7*d^3 - 840*b^4*d^4 + 1248*b^6*d^4 - 360*b^3*d^5 + 
     432*b^5*d^5 + 330*b^2*d^6 - 852*b^4*d^6 + 120*b*d^7 - 288*b^3*d^7 - 
     25*d^8 + 228*b^2*d^8 + 72*b*d^9 - 6*d^10, {0, 3}}, 
   {345*b^7 + 92*b^9 + 36*b^11 + 365*b^6*d + 54*b^8*d + 44*b^10*d - 
     570*b^5*d^2 - 324*b^7*d^2 - 132*b^9*d^2 - 710*b^4*d^3 - 180*b^6*d^3 - 
     176*b^8*d^3 + 105*b^3*d^4 + 420*b^5*d^4 + 168*b^7*d^4 + 325*b^2*d^5 + 
     216*b^4*d^5 + 264*b^6*d^5 + 120*b*d^6 - 236*b^3*d^6 - 72*b^5*d^6 + 
     20*d^7 - 108*b^2*d^7 - 176*b^4*d^7 + 48*b*d^8 - 12*b^3*d^8 + 18*d^9 + 
     44*b^2*d^9 + 12*b*d^10, {1, 1}}, 
   {800*b^7 - 1038*b^9 + 415*b^11 + 375*b^6*d - 427*b^8*d + 105*b^10*d - 
     1430*b^5*d^2 + 3138*b^7*d^2 - 1674*b^9*d^2 - 645*b^4*d^3 + 
     1222*b^6*d^3 - 414*b^8*d^3 + 460*b^3*d^4 - 3186*b^5*d^4 + 2546*b^7*d^4 + 
     165*b^2*d^5 - 1104*b^4*d^5 + 606*b^6*d^5 + 170*b*d^6 + 1110*b^3*d^6 - 
     1744*b^5*d^6 + 105*d^7 + 250*b^2*d^7 - 384*b^4*d^7 - 24*b*d^8 + 
     471*b^3*d^8 + 59*d^9 + 81*b^2*d^9 - 14*b*d^10 + 6*d^11, {0, 2}}, 
   {-345*b^6 + 138*b^8 + 72*b^10 + 52*b^12 - 585*b^5*d + 351*b^7*d + 
     26*b^9*d + 66*b^11*d + 120*b^4*d^2 + 195*b^6*d^2 - 258*b^8*d^2 - 
     190*b^10*d^2 + 555*b^3*d^3 - 546*b^5*d^3 - 78*b^7*d^3 - 264*b^9*d^3 + 
     210*b^2*d^4 - 840*b^4*d^4 + 342*b^6*d^4 + 240*b^8*d^4 + 30*b*d^5 + 
     39*b^3*d^5 + 78*b^5*d^5 + 396*b^7*d^5 + 15*d^6 + 543*b^2*d^6 - 
     198*b^4*d^6 - 100*b^6*d^6 + 156*b*d^7 - 26*b^3*d^7 - 264*b^5*d^7 - 
     36*d^8 + 42*b^2*d^8 - 20*b^4*d^8 + 66*b^3*d^9 + 18*b^2*d^10, {1, 0}}, 
   {-1820*b^6 + 2361*b^8 - 1232*b^10 + 342*b^12 - 510*b^5*d + 1509*b^7*d - 
     210*b^9*d + 88*b^11*d + 255*b^4*d^2 - 3702*b^6*d^2 + 3628*b^8*d^2 - 
     1302*b^10*d^2 - 315*b^3*d^3 - 2606*b^5*d^3 + 354*b^7*d^3 - 316*b^9*d^3 + 
     1685*b^2*d^4 + 339*b^4*d^4 - 3510*b^6*d^4 + 1788*b^8*d^4 + 825*b*d^5 + 
     685*b^3*d^5 + 198*b^5*d^5 + 384*b^7*d^5 - 120*d^6 + 984*b^2*d^6 + 
     1082*b^4*d^6 - 972*b^6*d^6 + 412*b*d^7 - 618*b^3*d^7 - 136*b^5*d^7 + 
     18*d^8 + 14*b^2*d^8 + 78*b^4*d^8 + 276*b*d^9 - 56*b^3*d^9 + 18*d^10 + 
     66*b^2*d^10 + 36*b*d^11, {0, 1}}, 
   {2040*b^5 - 3336*b^7 + 1400*b^9 - 486*b^11 + 96*b^13 - 420*b^4*d - 
     2248*b^6*d + 528*b^8*d + 16*b^10*d + 72*b^12*d + 3220*b^3*d^2 + 
     504*b^5*d^2 - 1788*b^7*d^2 + 1392*b^9*d^2 - 266*b^11*d^2 + 
     1740*b^2*d^3 + 1900*b^4*d^3 - 36*b^6*d^3 - 152*b^8*d^3 - 246*b^10*d^3 + 
     40*b*d^4 + 3216*b^3*d^4 - 398*b^5*d^4 - 1212*b^7*d^4 + 104*b^9*d^4 + 
     180*d^5 + 248*b^2*d^5 - 1578*b^4*d^5 + 360*b^6*d^5 + 264*b^8*d^5 - 
     384*b*d^6 + 560*b^3*d^6 + 144*b^5*d^6 + 324*b^7*d^6 + 100*d^7 + 
     1152*b^2*d^7 - 328*b^4*d^7 - 36*b^6*d^7 + 226*b*d^8 + 210*b^3*d^8 - 
     376*b^5*d^8 - 66*d^9 + 104*b^2*d^9 - 96*b^4*d^9 - 48*b*d^10 + 
     118*b^3*d^10 + 42*b^2*d^11, {0, 0}}}, OreAlgebraObject[{Der[d], Der[b]}, 
   Expand, #1 + #2 & , Expand[#1*#2] & , None], DegreeLexicographic]}
