Timeline for Is there any example of a Lie algebra which is not a derivation algebra?

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Dec 24, 2022 at 17:01	comment	added	Salvatore Siciliano		@Yves. Sure, this is a nice example as well!
Dec 22, 2022 at 10:18	comment	added	YCor		By the way, I should mention, in char $p>0$, the following 3-dimensional non-restrictable Lie algebra: for a nonzero scalar $c$, the Lie algebra $\mathfrak{g}(c)$ with basis $(X,Y,Z)$ and bracket $[X,Y]=Y$, $[X,Z]=cZ$, $[Y,Z]=0$. Then $\mathfrak{c}$ is restrictable if and only if $c^p=c$ (i.e., if and only if $c$ belongs to the prime subfield $\mathbf{F}_p$ (indeed otherwise $\mathrm{ad}(X)^p$ is not an inner derivation).
Nov 26, 2021 at 18:41	comment	added	Salvatore Siciliano		Consider the $(p+2)$-dimensional Lie algebra $L=Fx+Fy_1+\cdots+Fy_{p+1}$ with structure constants given by $[x,y_i]=y_{i+1}$ for $i=1,\ldots,p$, $[x,y_{p+1}]=0$, and $[y_i,y_j]=0$ for all $i,j=1,\ldots,p+1$. As $(\mathrm{ad}x)^p$ is not an inner derivation, it follows that $L$ cannot be restrictable.
Nov 26, 2021 at 15:09	comment	added	YCor		Nice. What's an example of non-restrictable Lie algebra (say, over an algebraically closed field of char $p>0$)?
Nov 26, 2021 at 14:53	history	edited	Salvatore Siciliano	CC BY-SA 4.0	added 1 character in body
Nov 26, 2021 at 10:59	history	answered	Salvatore Siciliano	CC BY-SA 4.0