  Let $\mathcal{O}$ be a Richardson nilpotent orbit in a simple Lie algebra
$\mathfrak{g}$ over $\mathbb C$, induced from a Levi subalgebra whose simple
roots are orthogonal short roots. The main result of the paper is a description
of a minimal set of generators of the ideal defining $\overline{ \mathcal{O}}$
in $S \mathfrak{g}^*$. In such cases, the ideal is generated by bases of at
most two copies of the representation whose highest weight is the dominant
short root, along with some fundamental invariants. This extends Broer's result
for the subregular nilpotent orbit. Along the way we give another proof of
Broer's result that $\overline{ \mathcal{O}}$ is normal. We also prove a result
connecting a property of invariants related to flat bases to the question of
when one copy of the adjoint representation is in the ideal in $S
\mathfrak{g}^*$ generated by another copy of the adjoint representation and the
fundamental invariants.
