Let $F$ be a field, with $\text{char}(F) = 0$, and $A \in M_{n\times n}(F)$ with $\text{tr}(A) = 0$. Show that there are matrices $B,C \in M_{n\times n}(F)$ that $A = BC - CB$.
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One nice proof is presented here – Ben Grossmann Jun 06 '16 at 17:56
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It also follows from this question, for the simple Lie algebra $\mathfrak{sl}_n(K)$ of traceless matrices. – Dietrich Burde Jun 06 '16 at 18:10
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Possible duplicate of Traceless matrices and commutators – Dietrich Burde Jun 06 '16 at 18:18