= Failure of Lie theorem in positive characteristic
In a <field> of <characteristic> $p$, take a <basis> $e_0,\ldots,e_{p-1}$, and let $xe_j=e_{j-1}$ with indices modulo $p$, while $ye_j=je_j$. Then $[x,y]=x$, so $kx+ky$ is a <Solvable Lie algebra>, but the distinct <eigenvalues> of $y$ force any common <eigenvector> to be a coordinate vector, and $x$ cyclically permutes those vectors. There is consequently no common <eigenvector>, even over an <algebraically closed field>.
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