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

This article describes a Lie subring property: a property that can be evaluated for a subring of a Lie ring
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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 $i\in \{1,2,\dots ,m\}$.

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
derivation-invariant Lie subring
ideal of a Lie ring