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If I understand 2.1 in Kac-Elashvili correctly, 2) is equivalent to $e$ being Richardson. Certainly, it implies $e$ is Richardson

This was wrong. If I recall correctly, there are principal Levi but non-Richardson nilpotentsyou really want to read a wrong answer, so those would all be counter exampleslook in the edit history.

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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.