  In a recent paper, Jason P. Bell and Jeffrey Shallit introduced the notion of
{\em Lie complexity} and proved that the Lie complexity function of an
automatic sequence is automatic. In this note, we give more facts concerning
Lie complexity and define the extended Lie complexity and the prefix Lie
complexity. Further, we prove that some proprieties of Lie complexity also hold
for the extended Lie complexity. Particularly, we prove that the extended Lie
complexity function and the first-order difference sequence of the prefix Lie
complexity function of an automatic sequence are both automatic.
