  We study the transverse Poisson structure to adjoint orbits in a complex
semi-simple Lie algebra. The problem is first reduced to the case of nilpotent
orbits. We prove then that in suitably chosen quasi-homogeneous coordinates the
quasi-degree of the transverse Poisson structure is -2. In the particular case
of {\emph subregular} nilpotent orbits we show that the structure may be
computed by means of a simple determinantal formula, involving the restriction
of the Chevalley invariants on the slice. In addition, using results of
Brieskorn and Slodowy, the Poisson structure is reduced to a three dimensional
Poisson bracket, intimately related to the simple rational singularity that
corresponds to the subregular orbit.
