  Bell and Shallit recently introduced the Lie complexity of an infinite word
$s$ as the function counting for each length the number of conjugacy classes of
words whose elements are all factors of $s$. They proved, using algebraic
techniques, that the Lie complexity is bounded above by the first difference of
the factor complexity plus one; hence, it is uniformly bounded for words with
linear factor complexity, and, in particular, it is at most 2 for Sturmian
words, which are precisely the words with factor complexity $n+1$ for every
$n$. In this note, we provide an elementary combinatorial proof of the result
of Bell and Shallit and give an exact formula for the Lie complexity of any
Sturmian word.
