SetDirectory["C:\\drorbn\\AcademicPensieve\\2012-12"];
<< FreeLie.m;
BCH[n_Integer] := BCH[⟨"x"⟩,⟨"y"⟩][n];
BCHWords[n_Integer] := Cases[BCH[n], _LW, Infinity];
AllLyndonWords[n_Integer] := AllLyndonWords[n, {"x", "y"}];

BCH[7]

Expand[2BCH[8]]

2BCH[8]-b[⟨"x"⟩, BCH[7]] // Expand

b[⟨"y"⟩, BCH[7]]// Expand

2BCH[10]-b[⟨"x"⟩, BCH[9]] // Expand

BCH[9]

b[⟨"x"⟩, #]& /@ BCHWords[9]

SymmetricDifference[A_List, B_List] := {
Complement[A,B], Complement[B,A]
};
Table[
SymmetricDifference[
b[⟨"x"⟩, #]& /@ BCHWords[2n-1],
BCHWords[2n]
],
{n,2,8}
] // ColumnForm