representation theory – Characterization of nilpotent elements even in $ mathfrak {sl} _n $

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