Lie subring invariant under any derivation with partial divided Leibniz condition powers

Definition
Suppose $$L$$ is a Lie ring and $$S$$ is a Lie subring of $$L$$. We say that $$S$$ is invariant under any derivation with partial divided Leibniz condition powers if the following holds: for any positive integer $$m$$ and any derivation with divided Leibniz condition powers up to $$m$$ for $$L$$ given by $$d^{(1)}, d^{(2)}, \dots, d^{(m)}$$, we have $$d^{(i)}(S) \subseteq S$$ for all $$i \in \{ 1,2,\dots,m\}$$.