# Real matrices commuting with quaternion matrices

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 $$M(4,\mathbb R)$$?