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A subspace of an algebra with involution is called a Lie skew-ideal if it isclosed under Lie products with skew-symmetric elements. Lie skew-ideals areclassified in central simple algebras with involution (there are eight of themfor involutions of the first kind and four for involutions of the second kind)and this classification result is used to characterize noncommutativepolynomials via their values in these algebras. As an application, we deducethat a polynomial is a sum of commutators and a polynomial identity of $d\timesd$ matrices if and only if all of its values in the algebra of $d\times d$matrices have zero trace.

Values of Noncommutative Polynomials, Lie Skew-Ideals and the Tracial Nullstellensatz