default order in FreeAlgebra

edit

In this link : http://doc.sagemath.org/html/en/refer... I am interested in the method lie_polynomial(w), which takes word from a FreeMonoid as input and give the output, the corresponding Lie monomial. But All these things depends on the order in the FreeAlgebra elements. Theoretically this is the lex order induced from the order on the alphabets, but in sage, the order in FreeAlgebra is it same as lex or different order?

F = FreeAlgebra(QQ, 3, 'x,y,z') 

M.<x,y,z> = FreeMonoid(3)

w = x*y*z

print F.lie_polynomial(w)

w = w.to_word()

w.lyndon_factorization()

This code giving me the output :

x*y*z - y*z*x

(xyz)

Since, w is Lyndon, the associated Lie monomial is [x [y z]] = xyz - xzy - yzx + zyx which is different from the above output, where am I making mistake. please tell.

The method of associating monomial to a word can be seen here : https://en.wikipedia.org/wiki/Free_Li...

Edit : There is a ticket regarding lie_polynomial( ) method. Please see here : https://trac.sagemath.org/ticket/22251

Thanks a lot once again.