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?



The non-simplicity of $SO(4)$ and $A_4$
Here are some general remarks which hold for arbitrary semisimple Lie algebras ${\mathfrak g}$ relating its algebraic properties to that of its Weyl group $W$. 


1. ${\mathfrak g}$ is simple if and only if the standard linear action of its Weyl group $W$ is irreducible. However, $W$ itself might still split nontrivially as a direct product (this happens in few cases); let's call such $W$ "reducible". This was analyzed by Luis Paris in <http://arxiv.org/pdf/math/0412214v2.pdf>. The main focus of his paper was on infinite Coxeter groups, but he also classified reducible finite Coxeter groups whose root system is irreducible (section 7). In all cases, the factor of $W$ by its center is irreducible, i.e., does not split nontrivially as a direct product. Therefore, the statement is: ${\mathfrak g}$ is simple iff $W/Z(W)$ is irreducible.
2. As for simplicity of $W$ itself, in the case of "classical" root systems, $W$ always has the form of semidirect product of a permutation group and a finite abelian group. Thus, if one is willing to divide by the abelian normal subgroup and then pass to the alternating group, then in the classical case one does obtains that simplicity of ${\mathfrak g}$ is equivalent to simplicity of a certain "subquotient" of $W$. I do not know enough about exceptional Coxeter groups to make a similar conclusion in general.


