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)?
Best Answer
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.