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Here's a positive answer of the question for arbitrary compact Lie groups. For a group $G$, denote $W_k(G)=\{x\in G^k:[x_1,\dots,x_k]=1\}$. Let $G$ be a compact Lie group. Then $W_k(G)$ has nonzero Haar measure for some $k\ge 1$ if and only if $G^0$ is a torus. One direction is obvious: if $G^0$ is a torus, then $(G^0)^k\subset W_k^G$ has positive measure (...


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