A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for involutions of the first kind and four for involutions of the second kind) and this classification result is used to characterize noncommutative polynomials via their values in these algebras. As an application, we deduce that a polynomial is a sum of commutators and a polynomial identity of d x d matrices if and only if all of its values in the algebra of d x d matrices have zero trace.
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机译：如果对合的代数子空间在具有对称对称元素的Lie乘积下是封闭的，则称其为Lie倾斜理想。李偏斜理想分类为具有对合的中心简单代数（其中有八种用于第一类对合，而第四类是第二种对合），该分类结果用于通过这些代数中的非交换多项式的值来表征。作为一个应用，我们推论一个多项式是换向子的总和，并且当且仅当它在d x d矩阵代数中的所有值都为零时才是d x d矩阵的多项式恒等式。
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