Products of commutators in simple algebras

Abstract

Let $A$ be a finite-dimensional simple algebra that is not a field. We show that every $a\in A$ can be written as $a=(bc-cb)(de-ed)$ for some $b,c,d,e\in A$. This is not always true for infinite-dimensional simple algebras. In fact, for any $m\in \mathbb{N}$ we provide an example of an infinite-dimensional simple unital $C^{\ast }$-algebra $A$ in which $1$ cannot be written as ${\sum }_{i=1}^m{x}_i(a_i{b}_i-b_i{a}_i)y_i$ for some $x_i,a_i,b_i,y_i\in A$.