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_] := x[x[AAA,BBB],CCC]-x[AAA,x[BBB,CCC]];

bd[AAA_,BBB_] := x[AAA,BBB]-x[BBB,AAA];


oo[AAA_,BBB_] := x[AAA,BBB]+x[BBB,AAA];


(*
[e,a]o(e,b,(e,c,d))
*)


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}];

AA = f[{z1,z2,z3,z4,z5,z6,z7,z8}];

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}];

BB = f[{z1,z2,z3,z4,z5,z6,z7,z8}];

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}];

CC = f[{z1,z2,z3,z4,z5,z6,z7,z8}];

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}];

DD = f[{z1,z2,z3,z4,z5,z6,z7,z8}];

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}];

EE = f[{z1,z2,z3,z4,z5,z6,z7,z8}];

EE = f[{e1,e2,e3,d4,e5,e6,e7,e8}];
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}];

(*
  [e,a]o(e,b,(e,c,d))
*)


g[AAA_,BBB_,CCC_,DDD_] := oo[ bd[EE,AAA],ass[EE,BBB,ass[EE,CCC,DDD]]];


ans =  Simplify[g[AA,BB,CC,DD]] ;
Print[ans];


ans =  Simplify[g[CC,DD,AA,BB]] ;
Print[ans];