In the first answer to: https://mathoverflow.net/questions/16026/the-finite-subgroups-of-sl2-c the correspondence between transverse slices of subregular elements of the nilpotent cone for a simple Lie algebra and du Val singularities is mentioned.
I'm curious, what happens if we start with a simple Lie algebra of a different type? (Are subregular elements well defined? wikipedia seems to indicate so but I cannot find a reference. As a side note, a reference for this fact would also be appreciated!)
I found an answer to this question. It was in Slodowy's "Simple singularities and simple groups" under the heading "inhomogeneous Dynkin diagrams." There's a table that gives all the cases but for example, a B_n type Lie algebra will have a A_{2n-1} type simple singularity, with a Z/2 of automorphisms. These automorphisms are due to the following: the group $G$ does not act on the slice but instead some subgroup of $G$; if the slice is given by $e + \ker(f)$, the intersection of the centralizers of $e$ and $f$ will. For types $ADE$ this is connected but for other types it is disconnected, and the finite group given by the different connected components gives an automorphism of a finite group on the slice.
I think the above is at least mostly correct but I am not sure, since I've just started reading on the material.