Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type   identities. I. Generalizations of the Capelli and Turnbull identities
  We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy-Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull's Capelli-type identities for symmetric and antisymmetric matrices. 
Sergio Caracciolo, Andrea Sportiello and Alan D. Sokal