Two results on centralisers of nilpotent elements

Let gg be a semisimple Lie algebra. We provide a short proof of McNinch’s result on centralisers of the sum of commuting nilpotent elements (J. Pure Appl. Algebra 206 (2006), 123–140) and characterise nilpotent elements e∈ge∈g having the property that the orbit G⋅eG⋅e is the largest nilpotent orbit meeting the centraliser of ee.