I am working through Stillwell's "Naive Lie Theory" and am completely stumped by the questions in this section. An example of one of the questions is
Show that the rotation that sends $1$ to $i$, $i$, to $-1$, and leaves $j,k$ fixed is the product of reflections in the hyperplanes orthogonal to $u_1=i$ and $u_2=(i-1)/\sqrt{2}$.
For the understanding of this issue, forget $j$ and $k$. This property can be explained by considering only the plane generated by $1$ and $i$, i.e., a copy of the complex plane, identifiable with $\mathbb{R^2}$. The rotation you mention is a $+\pi/2$ rotation that in fact can be written as the composition of 2 symmetries: the symmetry with respect to x-axis and the symmetry with respect to the axis with equation $y=x$ (it is a classical result that the composition of 2 symmetries with respect to axes making an angle $\theta$ is a rotation with angle $2 \theta$).
Edit: to make things clearer about the second symmetry. $(1-i)/\sqrt{2} \in \mathbb{C}$ corresponds to $(1/\sqrt{2},-1/\sqrt{2}) \in \mathbb{R}^2$ which is the unit normal vector to "hyperplane" with equation $(x/\sqrt{2}-y/\sqrt{2}=0$, i.e., $x=y$.