Agreed, but note (afaik) group theory is not usually applied in Control Theory, so it does seem it would be a challenge to someone in the field (maybe just a helpful disclaimer).
From a quick skim, it certainly looks like advanced mathematics is at use, but control theorists that deal with nonlinear systems have been using this kind of formalism already. Lie groups are even used in robotics to write down kinematics of serial chains.
I would counter that a minority of control engineers are using Lie groups and that a lot of people who have got by with linear methods so far might end up needing the conclusions in this book.