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Please verify whether my approach is right :

Let $I$ and $J$ be nilpotent ideals with nilpotent index $m$ and $n$ in a Lie algebra $L$. Using Aldo’s theorem, we can embed $ad L$ in some $gl_k$.

This means $ad I$ and $ad J$ consists nilpotent elements, so by Engels theorem both can embedded in $u_k$ which is the strict upper triangular matrices.

So $ad (I+J) $ is contained in $u_k$ which shows that $I+J$ is ad nilpotent. By Engels theorem, it follows that $I+J$ is nilpotent.

Are there any other approaches?

Thomas Andrews
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Melon_Musk
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    Using Aldo’s theorem - you mean, Ado's theorem. However, this is overkill, you don't need this here. The proof is elementary. – Dietrich Burde Feb 20 '25 at 13:20

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