# Characterisation of even nilpotent elements in $\mathfrak{sl}_n$

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).