If I understand 2.1 in Kac-Elashvili correctly, 2) is equivalent to $e$ being Richardson. Certainly, it implies $e$ is Richardson
If I understand 2.1 in Kac-Elashvili correctly, 2) is equivalent to $e$ being Richardson. Certainly, it implies $e$ is Richardson. If I recall correctly, there are principal Levi but non-Richardson nilpotents, so those would all be counter examples.