The (finite-dimensional) complex simple Lie algebras have been classified by Killing and Cartan a long time ago in the $\mathsf{A}_n,\mathsf{B}_n,\mathsf{C}_n,\mathsf{D}_n$ families and $\mathsf{G}_2,\mathsf{F}_4,\mathsf{E}_6,\mathsf{E}_7,\mathsf{E}_8$ exceptional cases. So the non-crystallographic $\mathsf{H}_3$ and $\mathsf{H}_4$ Coxeter groups (i.e., icosahedral symmetry) do not appear as Weyl groups of such algebras.

On the other hand, there is also no such thing as a field with one element, yet it is interesting to construct something which might play that role. Perhaps more relevantly here, there is no such thing as a tiling of the plane with regular pentagons, but there is something of the sort. And the concept of finite-dimensional simple Lie algebra can be generalized in various ways, e.g., Kac-Moody algebras.

So I've often wanted to ask: is there *something* which is remotely like the $\mathsf{H}_3$ and $\mathsf{H}_4$ Lie algebras or algebraic groups? They should probably be $33$- and $124$-dimensional respectively, and certainly the golden ratio $\frac{1+\sqrt{5}}{2}$ should play an important role in their definition (posing as an integer).