This is an interesting question from a class I'm taking that I'm not really sure how to approach.
It is known that we can define many embeddings of the quaternion algebra $\mathbb{H}$ in the real matrix algebra $M(4, \mathbb{R})$. The standard such embedding is defined by the map
$$ a + bi + cj + zk \mapsto \begin{bmatrix} a & -b & -c & -d \\ b & a & -d & c \\ c & d & a & -b \\ d & -c & b & a \\ \end{bmatrix} $$
Clearly, such matrices are invertible (such that their inverse is the image of the multiplicative inverse of the quaternion), satisfy $A^{T} = -A$, among other useful properties. If we define $\phi : \mathbb{H} \mapsto M(4, \mathbb{R})$ to be the above embedding, the problem is to describe the subspace $S \in M(4, \mathbb{R})$ of elements commuting with the image $\phi(\mathbb{H})$. That is,
$$S = \{A \in M(4, \mathbb{R}) : HAH^{-1} = A, \forall H \in \phi(\mathbb{H})\}$$
I can provide a few properties of such a subspace, but I'm not entirely sure how to classify it nicely. What can be said of such a subspace? Is there a way to view this subspace as a set of rotations or some other sort of geometric object?
With some work, one can write this subspace as $$\left\{\begin{pmatrix}a&b&c&d\\-b&a&-d&c\\-c&d&a&-b\\-d&-c&b&a\end{pmatrix}\colon a,b,c,d\in\mathbb R\right\}.$$ (The way I did this is to note that a matrix $M$ commutes with $\phi(\mathbb H)$ if and only if it commutes with $\phi(i)$ and $\phi(j)$, since such a matrix then commutes with $\phi(k)=\phi(ij)=\phi(i)\phi(j)$ and every linear combination thereof. I imagine there are plenty of other ways to do the algebra.) This looks an awful lot like the definition of $\phi$ -- a few negative signs are in different places, but it's pretty similar. Can you show that this is another embedding of $\mathbb H$ into $M(4,\mathbb R)$?