Is there a ''nice'' classification of **even** nilpotent elements in $\mathfrak{sl}_n,$ using the correspondence between nilpotent elements and partitions of n? By an even element, I mean an element $e$, such that for any (= every) $\mathfrak{sl}_2$-triple $(e,f,h)$ in $\mathfrak{sl}_n$, $ad(h)$ acts only with even eigenvalues.

By ''nice'' classification I mean to have an algorithm which takes a partition an tells whether (the coresponding element) is even or not.

Per Collingwood-McGovern (1993, Corollary 3.8.8) an element is even iff all labels on its weighted Dynkin diagram are 0 or 2. For partition $[d_1,\dots,d_k,0,\dots,0]$ these labels are the $h_i-h_{i+1}$, where $h_1\geqslant h_2\geqslant\dots\geqslant h_n$ is a reordering of $\bigcup_{i=1}^k\{d_i-1, d_i-3,\dots,-d_i+1\}$ (*ibid.*, Lemma 3.6.4).