Theorem 1. (Gerstenhaber, 1958): A vector space A of nilpotent (n x n)- matrices over a field with at least n elements cannot have dimension larger than n(n-1)/2, and if A has this maximal dimension, then it is conjugate to the space of nilpotent upper triangular matrices.

Recently, this theorem was generalised as follows:

Theorem 2. (Kuttler, Kraft, D. 2004): A subspace A of a complex semisimple Lie algebra g consisting entirely of nilpotent elements cannot have dimension larger than (dim(g)-rk(g))/2, and if A has this maximal dimension, then it conjugate to a maximal nilpotent subalgebra of g.

I will sketch our proof of Theorem 2 in the familiar setting of Theorem 1, and describe how these theorems fit into the larger project of finding sufficient conditions for subspaces of null-cones to be trivial, in an appropriate sense that generalises the conjugacy to the strictly upper triangular matrices in Theorem 1.