Ideal property is upper join-closed for Lie rings

This article gives the statement, and possibly proof, of a Lie subring property (i.e., ideal of a Lie ring) satisfying a Lie subring metaproperty (i.e., upper join-closed Lie subring property)
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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.

Related facts

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