  We prove that one of the conditions in M.V. Zaicev's formula for the
PI-exponent and in its natural generalization for the Hopf PI-exponent, can be
weakened. Using the modification of the formula, we prove that if a finite
dimensional semisimple Lie algebra acts by derivations on a finite dimensional
Lie algebra over a field of characteristic $0$, then the differential
PI-exponent coincides with the ordinary one. Analogously, the exponent of
polynomial $G$-identities of a finite dimensional Lie algebra with a rational
action of a connected reductive affine algebraic group $G$ by automorphisms,
coincides with the ordinary PI-exponent. In addition, we provide a simple
formula for the Hopf PI-exponent and prove the existence of the Hopf
PI-exponent itself for $H$-module Lie algebras whose solvable radical is
nilpotent, assuming only the $H$-invariance of the radical, i.e. under weaker
assumptions on the $H$-action, than in the general case. As a consequence, we
show that the analog of Amitsur's conjecture holds for $G$-codimensions of all
finite dimensional Lie $G$-algebras whose solvable radical is nilpotent, for an
arbitrary group $G$.
