  Let $G$ be a semisimple algebraic group with Lie algebra $\mathfrak g$. For a
nilpotent $G$-orbit $\mathcal O\subset\mathfrak g$, let $d_\mathcal O$ denote
the maximal dimension of a subspace $V\subset \mathfrak g$ that is contained in
the closure of $\mathcal O$. In this note, we prove that $d_\mathcal O \le
\frac{1}{2}\dim\mathcal O$ and this upper bound is attained if and only if
$\mathcal O$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim
V= \frac{1}{2}\dim\mathcal O$, then $V$ is the nilradical of a polarisation of
$\mathcal O$. Every nilpotent orbit closure has a distinguished $B$-stable
subspace constructed via an $\mathfrak{sl}_2$-triple, which is called the
Dynkin ideal. We then characterise the nilpotent orbits $\mathcal O$ such that
the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces
$\mathfrak c$ such that $\mathfrak c\cap\mathcal O$ is dense in $\mathfrak c$,
or (2) is the only $B$-stable subspace $\mathfrak c$ such that $\mathfrak
c\cap\mathcal O$ is dense in $\mathfrak c$.
