  We give a general definition of self-similar Lie algebras, and show that
important examples of Lie algebras fall into that class. We give sufficient
conditions for a self-similar Lie algebra to be nil, and prove in this manner
that the self-similar algebras associated with Grigorchuk's and Gupta-Sidki's
torsion groups are nil as well as self-similar. We derive the same results for
a class of examples constructed by Petrogradsky, Shestakov and Zelmanov.
