I think I have encountered a bug with the multiplication of coxeter group elements in the optional coxeter3 package. The following should return True but returns False:
from sage.libs.coxeter3.coxeter import CoxGroup as CoxGroup
W=CoxGroup(['A',2])
W([2,1])*W([2])==W([2,1,2])
It seems that elements are not properly put in normal form after multiplication:
W([2,1])*W([2])
returns
[2, 1, 2]
but
W([2,1,2])
returns
[1, 2, 1].
Sage version is 6.7.