  Properties of tensors equivalent to the direct product of two different
4-spinors are investigated. It is shown that the tensors obey additional 8
nonlinear restrictions, those are presented in Lorentz covariant form. In the
context of the Dirac-K\"{a}hler theory, such a property can be interpreted as
follows: if one wishes to consider the Dirac-K\"{a}hler field as consisting of
two 4-spinor fields, one must impose additional restrictions on tensors of the
Dirac-K\"{a}hler field, which leads to a non-linear wave equation for a complex
boson field (composed on the base of two 4-spinor fields). Instead, the use of
four bi-spinor fields gives possibility to construct the Dirac-K\"{a}hler
tensor set of 16 independent components. However, the formulas relating the
Dirac-K\"{a}hler boson to four fermion fields are completely different from
those previously used in the literature. In explicit form, restrictions on four
4-spinor corresponding to separation of different simplest bosons with spin 0
or 1 and various intrinsic parities, are constructed.
