  Let G be a connected reductive linear algebraic group defined over an
algebraically closed field of characteristic p. Assume that p is good for G. In
this note we consider particular classes of connected reductive subgroups H of
G and show that the cocharacters of H that are associated to a given nilpotent
element e in the Lie algebra of H are precisely the cocharacters of G
associated to e that take values in H. In particular, we show that this is the
case provided H is a connected reductive subgroup of G of maximal rank; this
answers a question posed by J.C. Jantzen.
