A normal form for admissible characters in the sense of Lynch

Parabolic subalgebras $\mathfrak{p}$ of semisimple Lie algebras define a ${\mathbb Z}$-grading of the Lie algebra. If there exists a nilpotent element in the first graded part of $\mathfrak{g}$on which the adjoint group of $\mathfrak{p}$ acts with a dense orbit, the parabolic subalgebra is said to be nice. The corresponding nilpotent element is also called admissible. Nice parabolic subalgebras of simple Lie algebras have been classified. In the case of Borel subalgebras a Richardson element of $\mathfrak{g}_1$ is exactly one that involves all simple root spaces. It is, however, difficult to write down such nilpotent elements for general parabolic subalgebras. In this paper we give an explicit construction of admissible elements in $\mathfrak{g}_1$ that uses as few root spaces as possible.