Fully invariant ideal of a Lie ring

Definition
Suppose $$L$$ is a Lie ring and $$S$$ is a subring of $$L$$. We say that $$S$$ is a fully invariant ideal of $$L$$ if $$S$$ is an ideal of $$L$$ and $$S$$ is a fully invariant Lie subring of $$L$$, i.e., it is invariant under all the Lie ring endomorphisms of $$L$$.

Facts

 * Fully invariant implies ideal for class two Lie ring