Ideal not implies derivation-invariant

Statement
There can exist a Lie ring $$L$$ and a subring $$I$$ of $$L$$ such that $$I$$ is an ideal of $$L$$ and is not a derivation-invariant subring of $$L$$.

Similar facts for Lie rings

 * Ideal property is not transitive for Lie rings
 * Characteristic not implies derivation-invariant
 * Characteristic ideal not implies derivation-invariant

Converse and related facts

 * Derivation-invariant subring implies ideal
 * Derivation-invariant subring of ideal implies ideal

Related facts for groups and other algebraic structures

 * Normal not implies characteristic

Proof
Let $$L$$ be an abelian Lie ring whose additive group is the Klein four-group. Thus, $$L = A \oplus B$$ where $$A,B$$ are subrings of size two. Since $$L$$ is abelian, both $$A$$ and $$B$$ are ideals of $$L$$.

Let $$\sigma$$ be the automorphism of $$L$$ that interchanges $$A$$ and $$B$$. Then, $$\sigma$$ is a derivation of $$L$$, but $$A$$ and $$B$$ are not invariant under $$\sigma$$.