Images of noncentral polynomials generate the entire simple algebra (with exceptions), and every noncommutative infinite simple algebra is generated by polynomial commutators.
Breˇ sar and P.ˇSemrl, The Waring problem for matrix algebras,Israel J
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On the structural behavior of images of polynomials
Images of noncentral polynomials generate the entire simple algebra (with exceptions), and every noncommutative infinite simple algebra is generated by polynomial commutators.