As in this paper On Prime Ideals of Lie Algebras
A Lie algebra $L$ satisfy the maximal condition for ideals, if for each , ascending chain $H_{1} \subseteq H_{2} \subseteq \ldots $ an index $m$ exists such that $H_{i}=H_{k}$ if $m<i$, $m<k.$ We say in short: $ L\in{\rm Max}-\triangleleft$.
An ideal $Q$ of $L$ is called semi-prime if $H^2 \subseteq Q$ with $H$ an ideal of $L$ implies $H\subseteq Q$.
$r(H)$ denote the intersection of all the prime ideals of $L$ containing $H$.
Corollary 8 from the paper State that: If $ L\in{\rm Max}-\triangleleft$, then $r(Q)=Q$ if and only if $Q$ is semiprime ideal.
My question:- If $ L \not \in{\rm Max}-\triangleleft$, Could someone give an example of a semiprime ideal $Q$ such that $r(Q) \neq Q$, please?
I think that Example 5.1 may be useful in this matter, but I hope that you help me to find semi-prime ideal $Q$ but $r(Q)\not=Q$. enter link description here