I read this recent question on proving that the identity is never the product of an odd number of reflections. After googling a bit, I found a Lemma in J. Aarts Plane and Solid Geometry, but I didn't fully understand the proof. I'll post the relevant sections, and the google book can be found here.
The Lemma that answers the linked question can be found on the google book, but I think I will understand it if I understand the proof of this above theorem.
I have two questions.
Why is it the case that if the three reflection axes are concurrent or all parallel does it follow that $\mathcal{F}$ is a reflection? For instance, I see in the case that if the axes are all parallel, then lines perpendicular to them will be invariant, but I don't necessarily see why there should be just one new axis of reflection that can replace the other three. I also played around with reflecting points across three concurrent axes, but didn't see why there should be one new reflection axis that "fits all" in a sense.
I'm uncomfortable with the line 'We can find $l_1^\prime$ and $l_2^\prime$ such that $S_{l_2^\prime}\circ S_{l_1^\prime}=S_{l_2}\circ S_{l_1}$ and $l^\prime_2$ is perpendicular to $l_3$. This seems like a lot to demand of these two lines, why should we be sure such lines exist?
Thanks for taking the time to read, it's been bugging me since I googled it when the other question was first posted.