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The Capelli identities claim $det(A)det(B) = det(AB+correction)$ for certainmatrices with noncommutative entries. They have applications in representationtheory and integrable systems. We propose new examples of these identities,constructed according to the following principle. For several known identitiesfor $n$ by $n$ matrices we construct new identity for $2n$ by $2n$ matriceswhere each element $z$ of the original matrix is substituted by 2x2 matrix ofthe form $[real(z) ~imag(z); ~ -imag(z) ~ real(z)]$, i.e. we view the originalidentity as complex valued, while the new identity is its real form(decomplexification). It appears that "decomplexification" affectsnon-trivially the "correction term". It becomes tridiagonal matrix, in contrastto the diagonal in the classical case. The key result is an extension to thenon-commutative setting of the fact that the determinant of the decomplexifiedmatrix is equal to the square module of the determinant of the original matrix(in non-commutative setting the corrections are necessary). The decomplexifiedCapelli's identities are corollaries of this fact and standard Capelliidentities. We also discuss analogs of the Cayley identity; observe that theCapelli determinant coincides with the Duflo image of the standard determinant;give short proof of the Cayley identity via Harish-Chandra's radial partcalculation.  The main motivation for us is a recent paper by An Huang (arXiv:1102.2657).From our viewpoint his result is a "dequaternionification" of 1 by 1 Capelliidentity. Apparently it can be extended to $n$ by $n$ case, but our approachshould be somehow modified for this.  The paper aims to be accessible and interesting not only for experts. Itgives brief review of Capelli identities, applications, their relation withWick quantization, Duflo map, some open issues, etc.

Decomplexification of the Capelli identities and holomorphic factorization