Given a simple Lie algebra over an algebraically closed field of good characteristic such as $\mathbb{C}$, its subvariety $\mathcal{N}$ of nilpotent elements has dimension $2N$ (where $N$ is the number of positive roots) and is the union of finitely many orbits under the adjoint group $G$. These orbits are partially ordered by $\mathcal{O'} \leq \mathcal{O}$ iff $\mathcal{O'} \subset \overline{\mathcal{O}}$. Some pictures are available here.

Using the Killing-Cartan classification of simple types $A$ - $G$, the orbits have been well studied. One general fact is that all orbits have even dimension, but for detailed results case-by-case work is usually essential. There is a unique dense regular orbit and a unique orbit just beneath it in the partial ordering: the subregular orbit, of dimension $2N-2$. Similarly, there is a unique minimal nonzero orbit (determined by any root vector for a long root), whose dimension was shown by Weiqiang Wang here to be $2 h^\vee -2$ with $h^\vee$ the dual Coxeter number of the root system: one greater than the sum of coefficients of the highest short root. Wang's proof uses standard facts about root systems and avoids the classification.

Meanwhile, Lusztig's refinement of Springer's work on Weyl group representations in the context of a desingularization of $\mathcal{N}$ led to a notion of special nilpotent orbit, not readily characterized within the Lie algebra setting. (But it has shown up independently in other active areas of representation theory.) The regular and zero orbits are always special; all Richardson orbits (including the subregular orbit) are special, but there are sometimes other special ones. There is always a unique minimal (nonzero) special orbit; it is determined by any root vector for a short root except in type $G_2$ where it is instead the subregular orbit.

The Lusztig-Spaltenstein duality map on the set of all orbits (generalizing the transpose map for partitions which parametrize orbits in type $A$) has as image the special ones, on which it induces a duality involution which interchanges the regular and zero orbits, etc. It turns out that only in types $A, D, E$ (where all roots have equal length) is the minimal nilpotent orbit special, thus in duality with the subregular orbit. Wang's result recently led me to observe case-by-case:

The minimal special nilpotent orbit has dimension $2h-2$.

Here $h$ is the Coxeter number (order of a Coxeter element in the Weyl group, or one greater than the sum of coefficients of the highest root). The formula seems not to be written down anywhere (?)

Is there a uniform proof of this dimension formula within the Lie algebra framework, avoiding the classification and using as little information as possible from Springer theory or other areas of representation theory?