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Let $F$ be a field which has a positive characteristic $p \ge 2$ and $(\mathfrak{g},[p])$ be a restricted Lie algebras over a field $F$ where $[p]$ is a $p$-th power map on $\mathfrak{g}$. $(\mathfrak{g},[p])$ is called simple restricted if $(\mathfrak{g},[p])$ has no non-trivial $p$-ideals (i.e. ideals $\mathfrak{h}$ of $(\mathfrak{g},[p])$ satisfies $x^{[p]} \in \mathfrak{h}$ for all $x \in \mathfrak{h}$), Similarly $(\mathfrak{g},[p])$ is called restricted simple if $\mathfrak{g}$ is simple as ordinary Lie algebra (more detailed definitions are found in R. Block and R. Wilson - Classification of the restricted simple Lie algebras (1988), pp. 116).

According to that paper restricted simple Lie algebras are simple restricted but its inverse is fails in general. I want to see an example that simple restricted Lie algebras but it is not restricted simple as possible as easy (e.g. low-dimension), moreover an explicit description of its non-trivial ideals. Thus my question is the following:

Question.1-(1): Find an example of a simple restricted Lie algebra $(\mathfrak{g},[p])$ that it is not restricted simple and satisfies the above conditions.

Question.1-(2): What is a non-trivial ideal of $(\mathfrak{g},[p])$ in Question.1-(1)?

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The simple restricted Lie algebras are exactly the minimal $p$-envelopes of the simple Lie algebras. In fact, if $(\mathfrak{g}, [p])$ is a simple restricted Lie algebra over a field $\mathbb{F}$ of characteristic $p>0$, then $[\mathfrak{g}, \mathfrak{g}]$ is simple as an ordinary Lie algebra and its minimal $p$-envelope is isomorphic to $\mathfrak{g}$. Conversely, if $L$ is a simple Lie algebra over $\mathbb{F}$, then its minimal $p$-envelope $\mathfrak{g}$ is a simple restricted Lie algebra. This gives rise to a one-to-one correspondence between simple restricted Lie algebras and simple (ordinary) Lie algebras. There are many references for this fact. For instance, this is nicely explained in Section 4 of Viviani - Simple finite groups and their infinitesimal deformations.

The simplest (and smallest) explicit example answering both the questions is given by the 3-dimensional Lie algebra $L=\mathbb{F}a+\mathbb{F}b+\mathbb{F}c$ with $[a,b]=c$, $[b,c]=a$ and $[c,a]=b$ over a field $\mathbb{F}$ of characteristic 2. In this case, $L$ is simple as an ordinary Lie algebra, and its minimal $p$-envelope $\mathfrak{g}=\mathbb{F}a+\mathbb{F}b+\mathbb{F}c+\mathbb{F}a^{[2]}+\mathbb{F}b^{[2]}$ is a simple restricted Lie algebra. Of course, $\mathfrak{g}$ is not simple as an ordinary Lie algebra, $[\mathfrak{g},\mathfrak{g}]$ being a non-trivial ideal.

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    $\begingroup$ Thank you for replying! Vivani's paper said that the above correspondence makes classification-theory of finite-dimensional modular Lie algebras of simple restricted ones. but, what is known that structures of minimal p-envelopes of simple modular Lie algebras in that classification? For example, how to calculate ideal in minimal p-envelope of Jacobson-Witt Lie algebra W(n,m)? If you know some references for this topics, please tell me. $\endgroup$ Jan 1, 2022 at 11:33
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    $\begingroup$ The best reference is the book "Simple Lie algebras over fields of positive characteristic. I. Structure theory" by Helmut Strade. In Theorem 7.2.2 of Chapter 7, you will find the description of the minimal p-envelopes of all finite-dimensional simple Lie algebras over algebracally closed fields of characteristic p>3. $\endgroup$ Jan 1, 2022 at 16:24
  • $\begingroup$ Thank you so much! As in the definition of p-envelopes, they extend from the original Cartan type Lie algebras a little (its proof seems to need many calculates which take so many times to follow that......). $\endgroup$ Jan 4, 2022 at 12:04

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