In theoretical physics, a Fierz identity is an identity that allows one to rewrite bilinears of the product of two spinors as a linear combination of products of the bilinears of the individual spinors. It is named after Swiss physicist Markus Fierz.
There is a version of the Fierz identities for Dirac spinors and there is another version for Weyl spinors. And there are versions for other dimensions besides 3+1 dimensions.
Spinor bilinears can be thought of as elements of a Clifford Algebra. Then the Fierz identity is the concrete realization of the relation to the exterior algebra. The identities for a generic scalar written as the contraction of two Dirac bilinears of the same type can be written with coefficients according to the following table.
For example, the V × V product can be expanded as,
Simplifications arise when the considered spinors are chiral or Majorana spinors as some term in the expansion can be vanishing.