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We know that when the killing form of a Lie algebra is nondegenerate then it is semisimple. I am looking for a semisimple Lie algebra with degenerate killing form. I know if the field is of characteristic zero it is impossible to find one.

Buddha
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In the exercises of Humphreys book on Lie algebras and Representation theory the example of $\mathfrak{sl}(3)$ (respectively of $\mathfrak{sl}(3)/Z(\mathfrak{sl}(3))$ in characteristic $3$ is worked out. Its Killing form is identical zero, but $\mathfrak{sl}(3)/Z(\mathfrak{sl}(3))$ is still a simple Lie algebra in characteristic $3$. One can find all details in the solutions here. The matrix of the Killing form relative to the standard basis is given by $$ \begin{pmatrix} 12 & -6 & 0 & 0 & 0 & 0 & 0 & 0\cr -6 & 12 & 0 & 0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 \cr 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 \cr 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 \end{pmatrix} $$ which is identical zero for $3=0$.

Dietrich Burde
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  • Thanks for the idea, but sorry you link seems to be redirecting me to an error page. Would it be possible to upload an alternative one? – asrxiiviii Oct 12 '20 at 09:43
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    @asrxiiviii Thank you for telling me about the link. I have updated the link. Searching "classical modular Lie algebra psl(n)" I obtain many other references. You can try these links by searching yourself . Then you are independent of outdated links. – Dietrich Burde Oct 12 '20 at 09:46