Lazard-divided Lie subring

Definition
Suppose $$L$$ is a defining ingredient::Lazard-divided Lie ring and $$S$$ is a subset of $$L$$. We say that $$S$$ is a Lazard-divided Lie subring of $$L$$ if $$S$$ is a Lie subring of $$L$$ that is closed under all the Lazard division operations, i.e., if $$x_1,x_2,\dots,x_p \in S$$, then $$t_p(x_1,x_2,\dots,x_p) \in S$$ as well.