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.
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机译：我们证明了由幂等元素组成的可归约李代数g的线性子空间的维数最多等于正根的数量，并且达到该上限的所有幂等子空间都等于Borel子代数g的nilradical。这概括了Gerstenhaber的经典定理，该定理针对n x n矩阵的代数陈述了这一事实。
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