Let $\mathbb H\cong \mathbb R^4$ be the quaternions with the Hamilton multiplication $\cdot \star\cdot :\mathbb H\times \mathbb H\to \mathbb H$ given by: \begin{align} (a\star b)_0=& a_0b_0-\sum_{j=1}^3a_jb_j\\ (a\star b)_i=& a_0b_i+a_i b_0 + \sum_{j,k=1}^3\epsilon_{ijk}a_jb_k, \end{align} for $a=(a_0,a_1,a_2,a_3)$ and $b=(b_0,b_1,b_2,b_3)$.
I want to find the period of the rotation $\exp(tc)\star x$, where $c=(0,c_1,c_2,c_3)$ is a pure quaternion and $x\neq 0$. With $i=(0,1,0,0)$ and $j=(0,0,1,0)$ and $k=(0,0,0,1)$ one can write: \begin{equation}\qquad\qquad \qquad \; \exp(tc)\star x=\exp(tic_1+tjc_2+tkc_3)\star x\,.\qquad\qquad \qquad\; (1)\end{equation}
So at first I thought that I have to solve for the lowest $t$ satisfying \begin{equation} \qquad \qquad \qquad \qquad tic_1+tjc_2+tkc_3\equiv 0\mod 2\pi\,,\qquad \qquad\qquad \qquad (2) \end{equation} which would be $2\pi \;lcm\left(\frac{1}{c_1},\frac{1}{c_2},\frac{1}{c_3}\right)=\frac{2\pi}{gcd(c_1,c_2,c_3)}$ whenever it is defined, and of course one has to treat the case where any one or two of the $c$ are zero, and the case where they are not rational, but still have a common multiple etc. In all the other cases there is no period.
But at a second glance I thought that (2) is not a necessary condition for (1) to hold, because after all $i,j$ and $k$ are quaternions and satisfy some algebra. So I could represent it as a matrix algebra and then I would have to use Baker-Campbell-Hausdorff formula or something, and get different results.
Is my first or second thought going in the right or wrong direction?
Exercise. The square roots of $-1$ inside $\mathbb{H}$ are precisely the pure imaginary unit quaternions.
Every imaginary unit quaternion is $\theta\mathbf{u}$ for a size $\theta$ and pure imaginary unit quaternion $\mathbf{u}$. Then de Moivre's formula states $\exp(\theta\mathbf{u})=\cos(\theta)+\sin(\theta)\mathbf{u}$. (Conversely every quaternion may be expressed $r\exp(\theta\mathbf{u})$ for some modulus $r>0$, angle $\theta$ and pure imaginary unit quaternion $\mathbf{u}$, so polar form exists for quaternions.)
In particular, this means for imaginary quaternions we have $\exp(\mathbf{u})=\exp(\mathbf{v})$ if and only if the vectors $\mathbf{u}$ and $\mathbf{v}$ are parallel (i.e. are both muliples of $\mathbf{t}$ for some unit pure imaginary quaternion $\mathbf{t}$) and differ by a vector of length $2\pi$ (i.e. $\mathbf{u}\equiv\mathbf{v}$ mod $2\pi\mathbf{t}$).