  The Capelli identities claim $det(A)det(B) = det(AB+correction)$ for certain
matrices with noncommutative entries. They have applications in representation
theory and integrable systems. We propose new examples of these identities,
constructed according to the following principle. For several known identities
for $n$ by $n$ matrices we construct new identity for $2n$ by $2n$ matrices
where each element $z$ of the original matrix is substituted by 2x2 matrix of
the form $[real(z) ~imag(z); ~ -imag(z) ~ real(z)]$, i.e. we view the original
identity as complex valued, while the new identity is its real form
(decomplexification). It appears that "decomplexification" affects
non-trivially the "correction term". It becomes tridiagonal matrix, in contrast
to the diagonal in the classical case. The key result is an extension to the
non-commutative setting of the fact that the determinant of the decomplexified
matrix is equal to the square module of the determinant of the original matrix
(in non-commutative setting the corrections are necessary). The decomplexified
Capelli's identities are corollaries of this fact and standard Capelli
identities. We also discuss analogs of the Cayley identity; observe that the
Capelli determinant coincides with the Duflo image of the standard determinant;
give short proof of the Cayley identity via Harish-Chandra's radial part
calculation.
  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 Capelli
identity. Apparently it can be extended to $n$ by $n$ case, but our approach
should be somehow modified for this.
  The paper aims to be accessible and interesting not only for experts. It
gives brief review of Capelli identities, applications, their relation with
Wick quantization, Duflo map, some open issues, etc.
