If $e$ is principal in a proper Levi subalgebra whose Dynkin diagram involves a component of type $A_k$ with $k$ odd, then e is not even. This is very easy to see by writing down an explicit $sl_2$-triple containing $e$. Occasionally, one can find another good grading for such $e$ (if it is Richardson) but this is quite rare outside type $A$. Also, if $e$ e\ne 0$is rigid in the sense of Lusztig-Spaltenstein then$e$is never even (nor Richardson) and for$g$exceptional all rigid nilpotent orbits except two (I think) are principal in Levi subalgebras. From the representation-theoretic viewpoint, rigid nilpotent elements give rise to the most interesting finite$W$-algebras. 2 added 124 characters in body; added 17 characters in body If$e$is principal in a proper Levi subalgebra whose Dynkin diagram involves a component of type$A_k$with$k$odd, then e is NOT not even. This is very easy to see by writing down an explicit$sl_2$-triple containing$e$. Occasionally, one can find another good grading for such$e$(if it is Richardson) but this is quite rare outside type$A$. Also, if$e$is rigid in the sense of Lusztig-Spaltenstein then$e$is never even (nor Richardson) and for$g$exceptional all rigid nilpotent orbits except two (I think) are principal in Levi subalgebras. From the representation-theoretic viewpoint, rigid nilpotent elements give rise to the most interesting finite$W$-algebras. 1 If$e$is principal in a proper Levi subalgebra whose Dynkin diagram involves a component of type$A_k$with$k$odd, then e is NOT even. This is very easy to see by writing down an explicit$sl_2$-triple containing$e$. Also, if$e$is rigid in the sense of Lusztig-Spaltenstein then$e$is never even and for$g$exceptional all rigid nilpotent orbits except two (I think) are principal in Levi subalgebras. From the representation-theoretic viewpoint, rigid nilpotent elements give rise to the most interesting finite$W\$-algebras.