TITLE: Semisimplicity of Lie algebra in positive characteristic
QUESTION [10 upvotes]: Let $F$ be a field of characteristic $p > 0$. Let $\mathfrak{g}$ be a linear Lie algebra, that is $\mathfrak{g}\subset M_n(F)$ for some natural number $n$. Does there exist a condition involving $n$ and $p$ such that $\mathfrak{g}$ is semisimple if and only if its Killing form is non-degenerate?

REPLY [9 votes]: If $g\subset M_n(F)$ and $n\le p-2$ then $g$ is semisimple if and only if the Killing form of g is non-degenerate. This statement is clear (and empty) if $p\in\{2,3\}$, while for $p>3$ one can use various results of the theory of modular Lie algebras. First one may assume that $g$ is semisimple as the converse statement is an easy exercise. One may also assume that $F$ is algebraically closed. Using Block's theorem one the structure of semisimple Lie algebras we then observe that $g$ is sandwiched between $\bigoplus_i S_i$ and $\bigoplus_i{\rm Der}(S_i)$ where the $S_i$'s are simple Lie algebras. As $n\le p-2$ the Classification Theorem implies that each $S_i$ is isomorphic to the Lie algebra of a simple algebraic $F$-group. Looking at the ranks of these groups more closely (and again using the inequality $n\le p-2$) we deduce that each $S_i$ has a non-degenerate Killing form and ${\rm Der}(S_i)=\mathrm{ad}\, S_i$ (this is discussed in Seligman's book quoted earlier). Hence the Killing form of $g\cong\bigoplus_i S_i$ is non-degenerate as well. The assumption on $p$ cannot be relaxed as for $p>3$ the Witt algebra $g={\rm Der}\,\mathcal{O}$ where $\mathcal{O}=F[X]/(X^p)$ is simple, has the zero Killing form, and acts irreducibly on the vector space $\mathcal{O}/F1$ of dimension $p-1$.