Can we make a suitable choice of y-axis in proving that any translation or rotation is the product of two reflections?

Question

In the book Geometry of Surface, John Stillwell proves that:

Any translation or rotation is the product of two reflections.

He starts with the translations. The proof starts with

By suitable choice of $y$-axis (namely, as a line parallel to the direction of translation) we can assume that the given translation is $t_{(0,\delta)}$.

I'm having trouble with this "suitable" choice, because in doing this we are actually doing a conjugation with a map that brings a line to the $y$-axis. And how do we know that this conjugation is still a product of two reflections? It seems like unsound reasoning to me. Can I get comments on this?

Here the translation $t_{(\alpha, \beta)}$ of $O$ to $(\alpha, \beta)$ is given by $x' = \alpha + x$ and $y' = \beta + y$.

Answer 1

The claim can be proven by simple euclidean geometry without introducing coordinates. I think he drew an auxiliary $y$-axis in order to explain the intended geometric construction.

Answer 2

If $\Pi$ is a reflection, then $O^T \Pi O$ is again a reflection, for any $O \in SO(n)$. (i.e. $O^T O = \mathbf{1}$ and $\det(O) > 0$

The composition of two reflection $\Pi_1 \circ \Pi_2$ can be written as $$ O^T \Pi_1 \Pi_2 O = O^T \Pi O O^T \Pi_2 O = \tilde{\Pi}_1 \tilde{\Pi}_2$$ Hence the conjugation is again the composition of two reflections $\tilde{\Pi}_i = O^T \Pi_i O$.

(I'm not an expert in this area. And as mentioned: a coordinate-free point of view is maybe better in this context.)