  Given a finite alphabet $\Sigma$ and a right-infinite word $\bf w$ over
$\Sigma$, we define the Lie complexity function $L_{\bf w}:\mathbb{N}\to
\mathbb{N}$, whose value at $n$ is the number of conjugacy classes (under
cyclic shift) of length-$n$ factors $x$ of $\bf w$ with the property that every
element of the conjugacy class appears in $\bf w$.
  We show that the Lie complexity function is uniformly bounded for words with
linear factor complexity, and as a result we show that words of linear factor
complexity have at most finitely many primitive factors $y$ with the property
that $y^n$ is again a factor for every $n$.
  We then look at automatic sequences and show that the Lie complexity function
of a $k$-automatic sequence is again $k$-automatic.
