f[{a_,b_,c_,d_}] := {
                      {a,{-b,-c,-d}},
                      {{b,c,d},a}
                    };

x[{{a_,B_},{C_,d_}},{{e_,F_},{G_,h_}}] := {
                                           {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;


A = {{a+b I,-{c-d I,e-f I ,g-h I}},
    {{c+d I,e+f I ,g+h I},a-b I}};


B = {{p+q I,-{r-s I,t-u I ,v-w I}},
     {{r+s I,t+u I ,v+w I},p-q I}};

alt = x[x[A,A],B] - x[A,x[A,B]];


X = {{a, {b,c,d}},
     {{e,f,g},h}};

Y = {{p,{q,r,s}},
     {{t,u,v},w}};

alt = x[x[X,X],Y]-x[X,x[X,Y]];


A = f[{a,b,c,d}];
B = f[{a,-b,-c,-d}];
X = f[{0,x,y,z}];

E0 = f[{1,0,0,0}];
E1 = f[{0,1,0,0}];
E2 = f[{0,0,1,0}];
E3 = f[{0,0,0,1}];

v0 = x[A,E0];
v1 = x[A,E1];
v2 = x[A,E2];
v3 = x[A,E3];

w0 = x[E0,A];
w1 = x[E1,A];
w2 = x[E2,A];
w3 = x[E3,A];


M =  {{ a,  b,  c,  d},
      {-b,  a,  d, -c},
      {-c, -d,  a,  b},
      {-d,  c, -b,  a}}



R  =  {{ a, b,  c,  d},
       {-b, a, -d,  c}, 
       {-c, d,  a, -b},
       {-d, -c, b,  a}};