# Sums of Commutators in a Group Ring

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?