cross[{a_,b_,c_},{d_,e_,f_}] := {0,b f - c e, -(a f-d c), a e - b d};
(*
    i   j   k
    a   b   c
    d   e   f
*)

x[{a_,b_,c_,d_},{e_,f_,g_,h_}]:={a e,0,0,0}+a{0,f,g,h}+{0,b,c,d}e+
cross[{b,c,d},{f,g,h}]+{- b f - c g - d h,0,0,0};


o = {1,0,0,0};
i = {0,1,0,0};
j = {0,0,1,0};
k = {0,0,0,1};