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For a single fermionic field, an interpretation of the Fierz identities(which establish relations between the bilinear field observables) is given.They appear closely related to the algebraic class (regular or singular) of thespin 2-form $S$ associated to the spinor field. If $S \neq 0$, the Fierzidentities follow from the 3+1 decomposition of the eigenvector equations for$S$ with respect to an inertial laboratory, which makes this interpretationsuitable for fermionic particle physics models. When $S= 0$, the Fierzidentities reduce to three constraints on the current densities associated withthe spinor field, saying that they are orthogonal, equimodular, the vectorcurrent being timelike and the axial one being spacelike.

Essential Fierz identities for a fermionic field