Two results on centralisers of nilpotent elements

Let image 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 image having the property that the orbit Gdot operatore is the largest nilpotent orbit meeting the centraliser of e.