Commutative reflections in the plane

Question

How can i show formally that two reflections $S_v ,S_w$ in the plane commute iff $v=±w$ or v and w are orthogonal. I tried and searched a lot, but did not manage to prove this.

Answer 1

$S_vv=-v$ and $v$ is the only such (unit) vector up to sign. Then

$$S_vS_wv=S_wS_vv=-S_wv$$ so either $S_wv=0$ or $S_wv=\pm v$ by the above.

If $S_wv=0$ then $v\perp w$ since eigenvectors of reflections are orthogonal.
If $S_wv=\pm v$ then $v=\pm w$, again by the above uniqueness, for $S_w$.