Nilpotent subspaces of maximal dimension in semisimple Lie algebras

We show that a linear subspace of a reductive Lie algebra g that consists of nilpotent elements has dimension at most equal to the number of positive roots, and that any nilpotent subspace attaining this upper bound is equal to the nilradical of a Borel subalgebra of g. This generalizes a classical theorem of Gerstenhaber which states this fact for the algebra of n x n matrices.