f[{a_,b_,c_,d_,e_,f_,g_,h_}] := { {a,{b,c,d}}, {{e,f,g},h} }; x[{{a_,B_},{C_,d_}},{{e_,F_},{G_,h_}}] := Simplify[{ {a e+B.G, a F + h B -Cross[C,G]}, {e C + d G + Cross[B,F], C.F + d h} }]; n[{{a_,B_},{C_,d_}}] := a d - B.C; xx[{{AA_,BB_},{CC_,DD_}},{{EE_,FF_},{GG_,HH_}}] := {{x[AA,EE]+x[BB,GG],x[AA,FF]+x[BB,HH]},{x[CC,EE]+x[DD,GG],x[CC,FF]+x[DD,HH]}}; ass[AAA_,BBB_,CCC_] := xx[xx[AAA,BBB],CCC]-xx[AAA,xx[BBB,CCC]]; la[AAA_,BBB_,CCC_] := ass[AAA,BBB,CCC]+ass[BBB,AAA,CCC]; ra[AAA_,BBB_,CCC_] := ass[AAA,BBB,CCC]+ass[AAA,CCC,BBB]; ls[AAA_,BBB_,CCC_] := ass[AAA,BBB,CCC]-ass[BBB,AAA,CCC]; rs[AAA_,BBB_,CCC_] := ass[AAA,BBB,CCC]-ass[AAA,CCC,BBB]; fl[AAA_,BBB_,CCC_] := ass[AAA,BBB,CCC]+ass[CCC,BBB,AAA]; AA = f[{a1,a2,a3,a4,a5,a6,a7,a8}]; BB = f[{b1,b2,b3,b4,b5,b6,b7,b8}]; CC = f[{c1,c2,c3,c4,c5,c6,c7,c8}]; DD = f[{d1,d2,d3,d4,d5,d6,d7,d8}]; EE = f[{e1,e2,e3,e4,e5,e6,e7,e8}]; FF = f[{f1,f2,f3,f4,f5,f6,f7,f8}]; GG = f[{g1,g2,g3,g4,g5,g6,g7,g8}]; HH = f[{h1,h2,h3,h4,h5,h6,h7,h8}]; PP = f[{p1,p2,p3,p4,p5,p6,p7,p8}]; QQ = f[{q1,q2,q3,q4,q5,q6,q7,q8}]; RR = f[{r1,r2,r3,r4,r5,r6,r7,r8}]; SS = f[{s1,s2,s3,s4,s5,s6,s7,s8}]; TT = f[{t1,t2,t3,t4,t5,t6,t7,t8}]; UU = f[{u1,u2,u3,u4,u5,u6,u7,u8}]; VV = f[{v1,v2,v3,v4,v5,v6,v7,v8}]; WW = f[{w1,w2,w3,w4,w5,w6,w7,w8}]; XX = f[{x1,x2,x3,x4,x5,x6,x7,x8}]; YY = f[{y1,y2,y3,y4,y5,y6,y7,y8}]; ZZ = f[{z1,z2,z3,z4,z5,z6,z7,z8}]; NN = f[{n1,n2,n3,n4,n5,n6,n7,n8}]; a1 = Random[Integer,{1,100}]; a2 = Random[Integer,{1,100}]; a3 = Random[Integer,{1,100}]; a4 = Random[Integer,{1,100}]; a5 = Random[Integer,{1,100}]; a6 = Random[Integer,{1,100}]; a7 = Random[Integer,{1,100}]; a8 = Random[Integer,{1,100}]; b1 = Random[Integer,{1,100}]; b2 = Random[Integer,{1,100}]; b3 = Random[Integer,{1,100}]; b4 = Random[Integer,{1,100}]; b5 = Random[Integer,{1,100}]; b6 = Random[Integer,{1,100}]; b7 = Random[Integer,{1,100}]; b8 = Random[Integer,{1,100}]; c1 = Random[Integer,{1,100}]; c2 = Random[Integer,{1,100}]; c3 = Random[Integer,{1,100}]; c4 = Random[Integer,{1,100}]; c5 = Random[Integer,{1,100}]; c6 = Random[Integer,{1,100}]; c7 = Random[Integer,{1,100}]; c8 = Random[Integer,{1,100}]; d1 = Random[Integer,{1,100}]; d2 = Random[Integer,{1,100}]; d3 = Random[Integer,{1,100}]; d4 = Random[Integer,{1,100}]; d5 = Random[Integer,{1,100}]; d6 = Random[Integer,{1,100}]; d7 = Random[Integer,{1,100}]; d8 = Random[Integer,{1,100}]; e1 = Random[Integer,{1,100}]; e2 = Random[Integer,{1,100}]; e3 = Random[Integer,{1,100}]; e4 = Random[Integer,{1,100}]; e5 = Random[Integer,{1,100}]; e6 = Random[Integer,{1,100}]; e7 = Random[Integer,{1,100}]; e8 = Random[Integer,{1,100}]; f1 = Random[Integer,{1,100}]; f2 = Random[Integer,{1,100}]; f3 = Random[Integer,{1,100}]; f4 = Random[Integer,{1,100}]; f5 = Random[Integer,{1,100}]; f6 = Random[Integer,{1,100}]; f7 = Random[Integer,{1,100}]; f8 = Random[Integer,{1,100}]; g1 = Random[Integer,{1,100}]; g2 = Random[Integer,{1,100}]; g3 = Random[Integer,{1,100}]; g4 = Random[Integer,{1,100}]; g5 = Random[Integer,{1,100}]; g6 = Random[Integer,{1,100}]; g7 = Random[Integer,{1,100}]; g8 = Random[Integer,{1,100}]; h1 = Random[Integer,{1,100}]; h2 = Random[Integer,{1,100}]; h3 = Random[Integer,{1,100}]; h4 = Random[Integer,{1,100}]; h5 = Random[Integer,{1,100}]; h6 = Random[Integer,{1,100}]; h7 = Random[Integer,{1,100}]; h8 = Random[Integer,{1,100}]; p1 = Random[Integer,{1,100}]; p2 = Random[Integer,{1,100}]; p3 = Random[Integer,{1,100}]; p4 = Random[Integer,{1,100}]; p5 = Random[Integer,{1,100}]; p6 = Random[Integer,{1,100}]; p7 = Random[Integer,{1,100}]; p8 = Random[Integer,{1,100}]; q1 = Random[Integer,{1,100}]; q2 = Random[Integer,{1,100}]; q3 = Random[Integer,{1,100}]; q4 = Random[Integer,{1,100}]; q5 = Random[Integer,{1,100}]; q6 = Random[Integer,{1,100}]; q7 = Random[Integer,{1,100}]; q8 = Random[Integer,{1,100}]; r1 = Random[Integer,{1,100}]; r2 = Random[Integer,{1,100}]; r3 = Random[Integer,{1,100}]; r4 = Random[Integer,{1,100}]; r5 = Random[Integer,{1,100}]; r6 = Random[Integer,{1,100}]; r7 = Random[Integer,{1,100}]; r8 = Random[Integer,{1,100}]; s1 = Random[Integer,{1,100}]; s2 = Random[Integer,{1,100}]; s3 = Random[Integer,{1,100}]; s4 = Random[Integer,{1,100}]; s5 = Random[Integer,{1,100}]; s6 = Random[Integer,{1,100}]; s7 = Random[Integer,{1,100}]; s8 = Random[Integer,{1,100}]; t1 = Random[Integer,{1,100}]; t2 = Random[Integer,{1,100}]; t3 = Random[Integer,{1,100}]; t4 = Random[Integer,{1,100}]; t5 = Random[Integer,{1,100}]; t6 = Random[Integer,{1,100}]; t7 = Random[Integer,{1,100}]; t8 = Random[Integer,{1,100}]; u1 = Random[Integer,{1,100}]; u2 = Random[Integer,{1,100}]; u3 = Random[Integer,{1,100}]; u4 = Random[Integer,{1,100}]; u5 = Random[Integer,{1,100}]; u6 = Random[Integer,{1,100}]; u7 = Random[Integer,{1,100}]; u8 = Random[Integer,{1,100}]; v1 = Random[Integer,{1,100}]; v2 = Random[Integer,{1,100}]; v3 = Random[Integer,{1,100}]; v4 = Random[Integer,{1,100}]; v5 = Random[Integer,{1,100}]; v6 = Random[Integer,{1,100}]; v7 = Random[Integer,{1,100}]; v8 = Random[Integer,{1,100}]; w1 = Random[Integer,{1,100}]; w2 = Random[Integer,{1,100}]; w3 = Random[Integer,{1,100}]; w4 = Random[Integer,{1,100}]; w5 = Random[Integer,{1,100}]; w6 = Random[Integer,{1,100}]; w7 = Random[Integer,{1,100}]; w8 = Random[Integer,{1,100}]; x1 = Random[Integer,{1,100}]; x2 = Random[Integer,{1,100}]; x3 = Random[Integer,{1,100}]; x4 = Random[Integer,{1,100}]; x5 = Random[Integer,{1,100}]; x6 = Random[Integer,{1,100}]; x7 = Random[Integer,{1,100}]; x8 = Random[Integer,{1,100}]; y1 = Random[Integer,{1,100}]; y2 = Random[Integer,{1,100}]; y3 = Random[Integer,{1,100}]; y4 = Random[Integer,{1,100}]; y5 = Random[Integer,{1,100}]; y6 = Random[Integer,{1,100}]; y7 = Random[Integer,{1,100}]; y8 = Random[Integer,{1,100}]; z1 = Random[Integer,{1,100}]; z2 = Random[Integer,{1,100}]; z3 = Random[Integer,{1,100}]; z4 = Random[Integer,{1,100}]; z5 = Random[Integer,{1,100}]; z6 = Random[Integer,{1,100}]; z7 = Random[Integer,{1,100}]; z8 = Random[Integer,{1,100}]; n1 = Random[Integer,{1,100}]; n2 = Random[Integer,{1,100}]; n3 = Random[Integer,{1,100}]; n4 = Random[Integer,{1,100}]; n5 = Random[Integer,{1,100}]; n6 = Random[Integer,{1,100}]; n7 = Random[Integer,{1,100}]; n8 = Random[Integer,{1,100}]; AAA = {{AA,BB},{CC,DD}}; BBB = {{EE,FF},{GG,HH}}; CCC = {{PP,QQ},{RR,SS}}; DDD = {{TT,UU},{VV,WW}}; EEE = {{XX,YY},{ZZ,NN}}; ans1 = ls[xx[AAA,AAA],AAA,AAA]; ans2 = rs[AAA,AAA,xx[AAA,AAA]]; ans3 = +ass[xx[AAA,AAA],AAA,AAA] -ass[AAA,AAA,xx[AAA,AAA]]; Clear[AA]; Clear[BB]; Clear[CC]; Clear[DD]; Clear[EE]; Clear[FF]; Clear[GG]; Clear[HH]; AA = f[{ a1, 0, 0, 0, 0, 0, 0, a1}]; BB = f[{ b1, b2, b3, b4, b5, b6, b7, b8}]; CC = f[{ b8,-b2,-b3,-b4,-b5,-b6,-b7, b1}]; DD = f[{ d1, 0, 0, 0, 0, 0, 0, d1}]; EE = f[{ e1, 0, 0, 0, 0, 0, 0, e1}]; FF = f[{ f1, f2, f3, f4, f5, f6, f7, f8}]; GG = f[{ f8,-f2,-f3,-f4,-f5,-f6,-f7, f1}]; HH = f[{ h1, 0, 0, 0, 0, 0, 0, h1}]; II = f[{ w1, 0, 0, 0, 0, 0, 0, w1}]; JJ = f[{ x1, x2, x3, x4, x5, x6, x7, x8}]; KK = f[{ x8,-x2,-x3,-x4,-x5,-x6,-x7, x1}]; LL = f[{ z1, 0, 0, 0, 0, 0, 0, z1}]; MM = f[{ p1, 0, 0, 0, 0, 0, 0, p1}]; NN = f[{ q1, q2, q3, q4, q5, q6, q7, q8}]; OO = f[{ q8,-q2,-q3,-q4,-q5,-q6,-q7, q1}]; PP = f[{ s1, 0, 0, 0, 0, 0, 0, s1}]; AAA = {{AA,BB},{CC,DD}}; BBB = {{EE,FF},{GG,HH}}; CCC = {{II,JJ},{KK,LL}}; DDD = {{MM,NN},{OO,PP}}; ans = Simplify[ xx[AAA,EEE] - xx[EEE,AAA] ]; o[AAA_,BBB_] := xx[AAA,BBB] + xx[BBB,AAA]; ja[AAA_,BBB_,CCC_] := o[o[AAA,BBB],CCC]-o[AAA,o[BBB,CCC]]; pp[AAA_,BBB_,CCC_,DDD_,EEE_] := o[AAA,o[BBB, o[CCC, o[DDD,EEE]]]]; P1 = pp[AAA,BBB,CCC,DDD,AAA]+ pp[AAA,CCC,DDD,BBB,AAA]+ pp[AAA,DDD,BBB,CCC,AAA]- pp[AAA,CCC,BBB,DDD,AAA]- pp[AAA,BBB,DDD,CCC,AAA]- pp[AAA,DDD,CCC,BBB,AAA]; P2 = pp[BBB,AAA,CCC,DDD,AAA]+ pp[CCC,AAA,DDD,BBB,AAA]+ pp[DDD,AAA,BBB,CCC,AAA]- pp[CCC,AAA,BBB,DDD,AAA]- pp[BBB,AAA,DDD,CCC,AAA]- pp[DDD,AAA,CCC,BBB,AAA]; P3 = pp[BBB,CCC,AAA,DDD,AAA]+ pp[CCC,DDD,AAA,BBB,AAA]+ pp[DDD,BBB,AAA,CCC,AAA]- pp[CCC,BBB,AAA,DDD,AAA]- pp[BBB,DDD,AAA,CCC,AAA]- pp[DDD,CCC,AAA,BBB,AAA]; P4 = pp[BBB,CCC,DDD,AAA,AAA]+ pp[CCC,DDD,BBB,AAA,AAA]+ pp[DDD,BBB,CCC,AAA,AAA]- pp[CCC,BBB,DDD,AAA,AAA]- pp[BBB,DDD,CCC,AAA,AAA]- pp[DDD,CCC,BBB,AAA,AAA]; Print[P1-P2]; Print[2 P1 - P3]; Print[2 P1 - P4];