It is well known that the alternating group $A_n$ is simple unless $n=4$. It is likewise well known that the special orthogonal group $SO(n)$ is essentially simple unless $n=4$ (specifically, the group $SO(n)$ is simple for odd $n$ and the group $SO(n)/\{\pm I\}$ is simple for even $n\neq 4$).

My question is: are these two facts equivalent? The non-simplicity of $SO(4)$ can be proved by observing that the double-cover of $SO(4)$ is $SU(2)\times SU(2)$ which, being a direct product, is very much **not** simple. This double-cover is closely related to properties of the quaternions (see Stillwell's *Naive Lie Theory*). Is there an analogous proof of the non-simplicity of $A_4$ based on a geometric structure related to the quaternions?

P.S. This relationship is an example of the "field of one element" heuristic. Can that be formalized?