Derivation with divided powers of a Lazard-divided Lie ring

Definition
Suppose $$L$$ is a Lazard-divided Lie ring. A derivation with divided powers for $$L$$ is defined as a defining ingredient::derivation with divided powers of the underlying Lie ring of $$L$$ that satisfies some additional compatibility conditions with respect to the Lazard division operations predicted by the multinomial theorem (analogous to the binomial formula for powers of a derivation).

Note that the degree one term must be a derivation of a Lazard-divided Lie ring for the Lazard-divided Lie ring $$L$$.