Let $ G$ be some group, and let $ \Bbb{C}G$ denote the complex group ring over $ G$ . Let $ x,y \in \Bbb{C}G$ , and define $ [x,y] := xy-yx$ to be a (ring) commutator in $ \Bbb{C}G$ . Let $ K$ be collection of all sums of commutators. I am reading a paper in which the authors claim that any element in $ K$ takes the form

$ $ \displaystyle \sum_{j,k} a_{jk} g_k^{-1} w_j g_k$ $

for some $ w_j , g_k \in G$ and $ a_{jk} \in \Bbb{C}$ with $ \sum_{k} a_{jk}=0$ for every $ j$ .

I tried proving this several times, but I always end up with some truly dreadful (triple) sums. See page 9 of this paper if you need more information. How does one go about proving the claim?