What interesting mathematical identities (or theorems) have been proven using quaternions?

Question

I've heard about the four square theorem and how it was proven using quaternions, which I found to be extremely fascinating, I was wondering whether there are any other interesting theorems which seem to have nothing to do with quaternions that were proven with quaternions, I looked up online but couldn't find any other than four square theorem, and there must be more. Thanks!

Answer 1

Theorem: There is a surjective group homomorphism from the group $SU(2)$ of all $2\times2$ complex unitary matrices with determinant $1$ onto the group $SO(3,\mathbb R)$ of all $3\times3$ orthogonal matrices with determinant $1$.

Sketch of proof: See $SU(2)$ as$$\left\{a+bi+cj+dk\in\mathbb H\,\middle|\,a^2+b^2+c^2+d^2=1\right\}.$$Let$$\operatorname{Im}\mathbb H=\left\{\alpha i+\beta j+\gamma k\,\middle|\,\alpha,\beta,\gamma\in\mathbb R\right\}.$$Then, for each $q\in SU(2)$ and each $r\in\operatorname{Im}\mathbb H$, $qrq^{-1}\in\operatorname{Im}\mathbb H$. It turns out that the linear map $r\mapsto qrq^{-1}$ has determinant $1$. Furthermore, it preserves the quaternionic norm. So, we can see this map as en element of $SO(3,\mathbb R)$ and this map from $SU(2)$ into $SO(3,\mathbb R)$ is actually surjective (and its kernel is $\pm\operatorname{Id}$).