On some properties of free commutators with semicircular variables

We investigate commutators of free variables of the form $i[x,s]$, where $s$ is a semicircular element. We show that although $s$ and $i[x,s]$ are not free, their sum nevertheless satisfies the free additive convolution identity $${\mu }_{s+i[x,s]}={\mu }_s⊞{\mu }_{i[x,s]}\mathrm{.}$$ Furthermore, we prove that the polynomial $x+i[x,s]$ is freely infinitely divisible whenever $x$ itself is freely infinitely divisible.