  Let $\g$ be simple Lie algebra. We give a conceptual proof for the fact that
the nilpotent orbits of height 3 are spherical. It is shown that if the highest
root of $\g$ is fundamental, then $\g$ has a specific nilpotent orbit of height
3. This orbit satisfies several interesting relations. Moreover, it can be used
for an intrinsic construction of a $G_2$ grading in $\g$. We also discuss an
approach to describing the algebra of covariants on a nilpotent orbit. For the
nilpotent orbits of height 2, it is shown that the algebra of regular functions
is a free module over the algebra of covariants. For the nilpotent orbits of
height 3, a conjectural description of the algebra of covariants is given. This
description is compatible with all previously known examples.
