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Let $L$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. Assume that the following condition holds:

For every restricted ideal $I$ of $L$, the minimal restricted subalgebras of $L/I$ are pairwise non-isomorphic.

QUESTION: Is $L$ necessarily abelian?

I already know that the answer is affirmative if one assumes that $L$ is nilpotent.

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