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$\DeclareMathOperator\ad{ad}$According to the Strade and Farnsteiners' textbook, a Lie algebra $L$ is restrictable if and only if $\ad(L)$ is $p$-subalgebra of $\operatorname{Der}(L)$. Also, the equivalent condition of this is explained that there is a map $[p]:L \to L$ such that $(L,[p])$ is a restricted Lie algebra. I want to prove these equivalent claims without the basis of Lie algebras. So, I want to define the map $[p]:L \to L$ directly without basis. Is it possible?

Memo: By Jacobson's theorem, if $L$ has a basis $(e_i)_{i \in \Lambda}$ ($\Lambda$ is an index set) and for every $e_i$ there is $y_i \in L$ such that $\ad^p_{e_i} = \ad_{y_i}$, $L$ has "unique" $p$-th power map $[p]$ such that $e^{[p]}_i = y_i$. This theorem is very useful to solve the problem, but it depends on basis of $L$.

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If $L$ is restrictable, then the $p$-map $[p]$ is not unique, and two $p$-maps on $L$ differ by a $p$-semilinear map $f:L \rightarrow Z(L)$, where $Z(L)$ is the center of $L$. Therefore, in order to define $[p]$ consinstently, you are forced to use the mentioned Jacobson's Theorem (and so a basis of $L$), unless $L$ is centerless. In the latter case, for every $x\in L$, the image $x^{[p]}$ under the $p$-map is just the unique $y\in L$ such that $ad(x)^{p}=ad(y)$.

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  • $\begingroup$ @Salvatore_Siciliano Thank you for replying always, and I'm sorry for being late to reply to your answer. So, To calculate $p$-map of given restricted Lie algebra $(L,[p])$, do we need to fix some basis of $L$ which is satisfied the conditions of that theorem? Does it depend on the choice of basis of $L$? $\endgroup$
    – Frank Voigt
    Commented Feb 19, 2023 at 7:45
  • $\begingroup$ A $p$-map is uniquely determined by the images of the elements of a basis. In general, for a Lie algebra, there could exist $p$-maps $[p]_1$ and $[p]_2$ such that $(L,[p]_1)$ and $(L,[p]_2)$ are not isomorphic as restricted Lie algebras. I suggest to have a look at the second chapter of the book by Strade and Farnsteiner. $\endgroup$
    – Salvatore Siciliano
    Commented Feb 19, 2023 at 11:14
  • $\begingroup$ That fact is affected by changing basis? What I want to tell you is, is it true $(L,[p]_1)$ and $(L,[p]_2)$ are isomorphic as restricted Lie algebras by changing the basis of $L$? $\endgroup$
    – Frank Voigt
    Commented Feb 25, 2023 at 6:55
  • $\begingroup$ In the first place, I am confused about the difference between the definition of Lie algebra using its basis and without basis:( $\endgroup$
    – Frank Voigt
    Commented Feb 25, 2023 at 7:01

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