If $G$ is a group with a supersimple theory having a finite $SU$-rank, then the subgroup of $G$ generated by all of its normal nilpotent subgroups is definable and nilpotent. This answers a question asked by Elwes, Jaligot, Macpherson and Ryten. If $H$ is any group with a supersimple theory, then the subgroup of $H$ generated by all of its normal soluble subgroups is definable and soluble.
"On the definability of radicals in supersimple groups." J. Symbolic Logic 78 (2) 649 - 656, June 2013. https://doi.org/10.2178/jsl.7802160