The theoretical background of subsystem TDDFT (sTDDFT) has been reviewed both in our original article and in a perspective article focussing on this topic , which also discusses recent applications of sTDDFT. Here, we briefly repeat some of the basic ideas and working equations with a special focus on recent developments, mainly following the presentation in Refs. For the sake of brevity, we will omit spin labels in the following; the full formalism explicitly considering spin labels is given in Ref. , and an earlier derivation of the unrestricted formalism (neglecting environmental response, however) can be found in Ref. .