Generalizations
Much work have been done on the identity and its generalizations. Approximately two dozens of mathematicians and physicists contributed to the subject, to name a few: R. Howe, B. Kostant Fields medalist A. Okounkov A. Sokal, D. Zeilberger.
It seems historically the first generalizations were obtained by Herbert Westren Turnbull in 1948, who found the generalization for the case of symmetric matrices (see for modern treatments).
The other generalizations can be divided into several patterns. Most of them are based on the Lie algebra point of view. Such generalizations consist of changing Lie algebra to simple Lie algebras and their super (q), and current versions. As well as identity can be generalized for different reductive dual pairs. And finally one can consider not only the determinant of the matrix E, but its permanent, trace of its powers and immanants. Let us mention few more papers; still the list of references is incomplete. It has been believed for quite a long time that the identity is intimately related with semi-simple Lie algebras. Surprisingly a new purely algebraic generalization of the identity have been found in 2008 by S. Caracciolo, A. Sportiello, A. D. Sokal which has nothing to do with any Lie algebras.
Read more about this topic: Capelli's Identity