I know $\mathfrak{sl}(n,k)$ isn't solvable if char$k\ne2$, and also $\mathfrak{sl}(2,k)$ is nilpotent if char$k=2$. What about $\mathfrak{sl}(n,k)$ when char$k=2$ in general?
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The Lie algebra $\mathfrak{sl}_n(K)$ for $n>1$ is simple over any field of characteristic $p$ if and only if $p\nmid n$. In case of $p\mid n$, the center is nontrivial, but the quotient then by its center is again simple for $n>2$. This "new" simple Lie algebra is denoted by $\mathfrak{psl}_n(K)$ for $p\mid n$.
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Dietrich Burde
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What is the relation between simple and solvable/nilpotent? – roob Jun 07 '22 at 08:40
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@DietrichBurde I think you mean the other way round: nilpotent implies solvable but not the converse – Callum Jun 07 '22 at 13:59
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1By definition, a simple Lie algebra is not solvable (and hence not nilpotent), and vice versa. See also this post and others at this site. – Dietrich Burde Jun 07 '22 at 14:01