In my lectures the following was said:
But I have no idea how to take the tangent map of (3.19) to get that expression. More generally, I realized I don't know how to take tangent maps in complicated scenarios and examples/explanations would be very helpful!
Edit:
I know that $T_e(T_eF\circ Ad)=T_{Ad(e)}T_eF \circ T_e Ad=T_eF\circ ad$ but when I attempt this with the RHS $T_e(Ad(F)\circ T_eF)=T_{F(e)}(Ad\circ F) T_e T_eF$ clearly doesn't work (the $T$ aren't even acting on the right things) but I have no idea what to do instead.