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But Rudin's proof does not work for arbitrary complete fields -- he uses the fact that the unit ball in $\mathbb C^n$ is compact, so his argument will not work if $K$ is a complete but not locally compact field -- for example, the completion of the algebraic closure of $\mathbb Q_p$.
@VictorProtsak: to play devil's advocate, to some extent that depends on how you state the theorem -- if $(V,\rho)$ is a $\mathfrak g$-representation, and we write $t_V(x,y) = \text{tr}(\rho(x)\rho(y))$, then $\rho(\mathfrak g)$ is solvable if and only if $t_V$ vanishes on $D(\mathfrak g)$. This obviously implies Cartan's Criterion, and it is also potentially more useful in the sense that if a Lie algebra is given as a subalgebra of some $\mathfrak{gl}(V)$, the restriction of the trace form may be simpler to use that the Killing form.
