Self-derivation-invariant Lie subring

Definition
A Lie subring $$A$$ of a Lie ring $$L$$ is termed a self-derivation-invariant Lie subring if the following holds. Consider a map $$d:A \to L$$ that is a derivation from the Lie subring to the whole Lie ring viewed as a module over it. Then, $$d(A) \subseteq A$$.

Weaker properties

 * Stronger than::Derivation-invariant Lie subring
 * Stronger than::Ideal of a Lie ring

Facts

 * Homomorph-containing subgroup of additive group of Lie ring is self-derivation-invariant and homomorph-containing