  Let g be a simple Lie algebra, with fixed Borel subalgebra b and with Weyl
group W. Expanding on previous work of Fan and Stembridge in the simply laced
case, this note aims to study the fully commutative elements of W, and their
connections with the spherical nilpotent orbits in g. If g is not of type G_2,
it is shown that an element w in W is fully commutative if and only if the
subalgebra of b determined by the inversions of w lies in the closure of a
spherical nilpotent orbit. A similar characterization is also given for the
ad-nilpotent ideals of b, which are parametrized by suitable elements in the
affine Weyl group of g thanks to the work of Cellini and Papi.
