Determinant characters and multiplicative solubility of finite skew braces

Let $B$ be a finite skew brace. We prove that if the additive derived subgroup $(B,+{)}^{\mathrm{\prime }}$ is nilpotent, then the multiplicative group $(B,\cdot )$ is soluble. In particular, this holds for finite skew braces with supersoluble or metabelian additive group.