Lie groups come up very naturally in synthetic differential geometry (SDG) (http://www.fuw.edu.pl/~kostecki/sdg.pdf http://www.fuw.edu.pl/~kostecki/sdg.pdf). In

Lie groups come up very naturally in synthetic differential geometry (SDG) (http://www.fuw.edu.pl/~kostecki/sdg.pdf http://www.fuw.edu.pl/~kostecki/sdg.pdf). In fact, the main advantage of SDG over the classical formulation of differential geometry is that it can be developed in a largely coordinate free fashion and that you can talk natively about differential forms. That said, there is a design space here and while SDG is close enough to ordinary mathematics that you can explain it to high-school students, it might not be the best fit for developing modern physics. There are several different approaches (e.g., working in the internal language of a "synthetic differential infinity-topos") which seem promising. It's an area of active research and one that is still extraordinarily difficult to get into.