Balanced factorisations in some algebras
  We prove that, in any field of characteristic not two and not three except the five-element field, each element decomposes into a product of four factors whose sum vanishes. We also find all $k,n,q$ such that every $n\times n$ matrix over the $q$-element field decomposes into a product of $k$ commuting matrices whose sum vanishes. 
Anton A. Klyachko, Andrey M. Mazhuga, Anastasia N. Ponfilenko
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