Assume that $A$ is a unital $C^{*}$-algebra. Let $a\in A$ be a nilpotent element with $$a^{k}=0,\;\;k>1.$$
Are there two elements $x,y\in A$ with $a=xy,\;\;(yx)^{k-1}=0$?
Motivation for this question is described here.
Note: The above property can be considered as a ring theoretical invariant. It would be interesting to classify all unital rings with this property.