= Conjugate-spectrum proof of Cartan solvability
{c}
{title2=$\operatorname{tr}(xy)=\sum_\lambda\dim(V_\lambda)|\lambda|^2$}
For $x$ in the <derived algebra> of a complex matrix <Lie algebra> $L$, define $y$ by multiplication by $\overline\lambda$ on each <generalized eigenspace> $V_\lambda$ of $x$. <Polynomial interpolation> and <adjoint compatibility of additive Jordan decomposition> make $\operatorname{ad}y$ a <polynomial> in $\operatorname{ad}x$ with zero constant term, so $[y,L]\subseteq[L,L]$. Trace orthogonality of $[L,L]$ and $L$ forces the displayed sum to vanish. Thus $x$ is a <nilpotent endomorphism>, and the <Engel theorem> implies solvability. The auxiliary $y$ and the Jordan components are not required to lie in $L$.
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