Ideal property is upper join-closed for Lie rings

Statement
Suppose $$L$$ is a Lie ring and $$I$$ is a subring of $$L$$. Suppose $$A_j$$ are subrings of $$L$$ both containing $$I$$. Then, if $$I$$ is an ideal of each of the $$A_j$$s, $$I$$ is also an ideal of the Lie subring of $$L$$ generated by the $$A_j$$s.

Analogues in other algebraic structures
Similar facts in other algebraic structures:


 * Normality is upper join-closed (for groups)
 * Ideal property is upper join-closed for associative rings

Opposites in other algebraic structures:
 * Ideal property is not upper join-closed for alternating rings
 * Normality is not upper join-closed for algebra loops