There is an isomorphism between quaternions and $4\times 4$ matrices: $$ \phi: a+bi+cj+dk \longmapsto \begin{pmatrix} a&b&c&d \\ -b&a&-d&c\\ -c&d&a&-b\\ -d&-c&b&a \end{pmatrix} $$ Matrices of such kind have a peculiar eigendecomposition. To be more specific, they have $2$ eigenvalues $\lambda = a \pm i \sqrt{b^2+c^2+d^2}$, each with a $2$-dimensional eigenspace spanned on $$ e_1 = \begin{pmatrix}b \\ \pm i\sqrt{b^2+c^2+d^2} \\ d \\ -c\end{pmatrix}, e_2 = \begin{pmatrix}\pm i\sqrt{b^2+c^2+d^2} \\ b \\ c \\ d\end{pmatrix} $$
Q1: What's actually going on? Is there a simple and direct way to explain this?
Q2: Is it possible to generalize this result to know the eigendecomposition of the stuff that you get in the Cayley-Dickson procedure beyond quaternions? Even without an isomorphism, you still can represent a multiplication by a particular hypercomplex number with a matrix, so I'd be curious in its eigendecomposition.
UPD: Some motivation behind the question is that if we ignore the signs, we get a much simpler eigendecomposition:
$$ \begin{pmatrix} a&b&c&d \\ b&a&d&c\\ c&d&a&b\\ d&c&b&a \end{pmatrix} = \frac{1}{4}H_4 \operatorname{diag}\begin{pmatrix} a+b+c+d\\ a-b+c-d\\ a+b-c-d\\ a-b-c+d \end{pmatrix} H_4, $$
where $H_4$ is the Walsh matrix:
$$ H_4 = \begin{pmatrix} 1&1&1&1 \\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1 \end{pmatrix}. $$
Knowing this, I was hoping for an interesting structure here as well.
