Pensieve header: Fuller output of for the direct KV solution. Even fuller: <a href=KVDirect-to-13.m>KVDirect-to-13.m</a>.

SetDirectory["~drorbn/AcademicPensieve/Projects/WKO4"];
<< FreeLie.m;
<< AwCalculus.m;
x=LW@"x"; y=LW@"y"; u=LW@"u";
{F=LS[{x,y}, Fs], G=LS[{x,y}, Gs]}; Fs["y"]=1/2;
SeriesSolve[{F,G},
(ℏ^(-1))(LS[x+y]-BCH[y,x]≡F-G-Ad[-x][F]+Ad[y][G])∧(Subscript[div, x])[F]+(Subscript[div, y])[G]≡(1/2)(Subscript[tr, u])[adSeries[(ad/((E^ad)-1)), x][u]+adSeries[(ad/((E^ad)-1)), y][u]-adSeries[(ad/((E^ad)-1)), BCH[x,y]][u]]];
{F,G}

{F@{8},G@{8}}  // Timing

The same thing, copy-paste ready and machine readable:

Sum[{F[k], G[k]}, {k,8}] // InputForm

The same thing, machine readable and to degree 13, is in the file quoted below, in the same folder:

Put[Sum[{F[k], G[k]}, {k,13}], "KVDirect-to-13.m"];

{TimeUsed[], MaxMemoryUsed[]}