TITLE: Example of a semisimple Lie algebra with degenerate Killing form
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: 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 $3=0$.