2-sub-ideal of a Lie ring

Definition
A subring of a Lie ring is termed a 2-sub-ideal if it is an ideal of an ideal of the Lie ring.

Lie algebra
A 2-sub-ideal of a Lie algebra is a 2-sub-ideal of a Lie ring that is also a linear subspace.

Note that there is a somewhat subtle but important point here. For the definition of 2-sub-ideal, we need it to be an ideal of an ideal of the whole ring. It turns out that, if the bottom subset is a linear subspace of the Lie algebra, we can choose the intermediate ideal to also be a linear subspace, and hence to be an ideal in the Lie algebra sense.

Group via the Lazard correspondence
Suppose $$G$$ is a Lazard Lie group and $$L$$ is its Lazard Lie ring. Under the natural bijection between $$L$$ and $$G$$, the 2-sub-ideals of $$L$$ correspond to the 2-subnormal subgroups of $$G$$.

Stronger properties

 * Weaker than::Ideal of a Lie ring

Weaker properties

 * Stronger than::Sub-ideal of a Lie ring