let $q =\cos\frac{\theta}{2}$ + $\sin\frac{\theta}{2}$i
and consider the map $x \to qxq^{-1} + i + j$
why is it a helical displacement? the axis of rotation is $i$, but the translation is $i + j$, and i thought the translation must parallel to the axis of rotation.
and how could i find the Chasles decomposition of this transformation?
On the $jk$-plane (i.e., orthogonal complement to our axis $\langle i\rangle$), rotation about by $\theta$ then translate by $j$ fixes the point $\frac12j+\frac{\sin\theta}{2-2\cos\theta}k$. So you can describe the map as "translate by $i$ and rotate by $\theta$ about the axis $\frac12j+\frac{\sin\theta}{2-2\cos\theta}k+\langle i\rangle$.