  The condition of nilpotency is studied in the general linear Lie algebra
$\mathfrak{gl}_{n}(\mathbb{K})$ and the symplectic Lie algebra
$\mathfrak{sp}_{2m}(\mathbb{K})$ over an algebraically closed field of
characteristic 0. In particular, the conjugacy class of nilpotent matrices is
described through nilpotent orbit varieties $\mathcal{O}_{\lambda}$ and an
algorithm is provided for computing the closure
$\overline{\mathcal{O}_{\lambda}} \cong
\text{Spec}\left(\mathbb{K}[X]\big/J_{\lambda}\right).$ We provide new
generators for the ideal $J_{\lambda}$ defining the affine variety
$\overline{\mathcal{O}_{\lambda}}$ which show that the generators provided in
[J.Weyman - "The equations of conjugacy classes of nilpotent matrices", 1989]
are not minimal. Furthermore, we conjecture the existence of local weak
N\'{e}ron models for nilpotent orbit varieties based on bounding $p$ in the
polynomial ring with p-adic integer coefficients for which the equations
defining $\mathcal{O}_{\lambda}$ can embed.
