Suppose there are two elements $x,y\in \mathfrak{gl}(V)$ for $V$ finite dimensional vector space over an algebraically closed field. If both $x,y$ commute with $z=[x,y]$, then how to show that $z$ is nilpotent?
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Presumably you meant to also assume that the field has characteristic $0$? Because otherwise it does not hold. – Tobias Kildetoft Nov 01 '20 at 18:49
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yes, you are right! – Christina Nov 01 '20 at 18:52
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Hint: take an eigenspace of $z$ with respect to a nonzero eigenvalue. – YCor Nov 01 '20 at 20:24
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1Or use Lie's theorem, see here. – Dietrich Burde Nov 01 '20 at 20:24