Great! The reference to Bonnafe I believe does say that I got all the examples. It is of course clear that the elements s that I was looking at had a centralizer whose connected component must be a semi-simple group containing the maximal torus, so I was in what's called the Borel-Siebenthal situation with the extra knowledge that the Weyl group of C(s) is W. I knew among classical groups this was possible only for O(2n) < SO(2n+1). However, I was not sure on the Exceptional groups. The paper of Bonnafe I believe shows this phenomenon does not happen for exceptional groups. Thank you!

There was a 1 in the (n+1)st place!

The action of the Weyl group for PGL(2) is t--->t^{-1} so indeed -1 is left invariant under the Weyl group?