### default order in FreeAlgebra

In this link : http://doc.sagemath.org/html/en/reference/algebras/sage/algebras/free_algebra.html 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?

Thanks for your valuable timing.

Have a good day.

Edit :

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

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

w = x*y*~~z
~~z

print ~~F.lie_polynomial(w)
~~F.lie_polynomial(w)

w = ~~w.to_word()
~~w.to_word()

w.lyndon_factorization()

This code giving me the out output ~~:
~~:

x*y*z - y*z*~~x
~~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.

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

Thanks a lot once again.