how to prove the Lemma 5.6. of "Invitation to 3D vision"

Question

In the proof of Lemma 5.6 on page 115, there are some points which I am not quite clear, as shown in the figure:enter image description here

I don't quite understand the proof ,in the line part of the figure. Concretely about How to deduce the formula 5.8 in detail

Answer 1

You already know $R=e^{\hat{\omega}\theta}$, and the proof has just shown that $\omega = \pm \frac{T}{||T||}$. Hence $R=\exp\left(\pm\hat{T}\frac{\theta}{||T||}\right)$. Then since for any matrix $A$ and constant $c$, $e^{cA}$ commutes with $A$, it follows that $R$ commutes with $\hat{T}$. So (5.7) becomes $R^2\hat{T} = \hat{T}$, which gives (5.8).