  It is known that in the four-dimensional Riemannian space the complex
bispinor generates a number of tensors: scalar, pseudo-scalar, vector,
pseudo-vector, antisymmetric tensor. This paper solves the inverse problem: the
above tensors are arbitrarily given, it is necessary to find a bispinor
(bispinors) reproducing the tensors. The algorithm for this mapping constitutes
construction of Hermitean matrix $M$ from the tensors and finding its
eigenvalue spectrum. A solution to the inverse problem exists only when $M$ is
nonnegatively definite. Under this condition a matrix $Z$ satisfying equation
$M=ZZ^{+}$ can be found. One and the same system of tensor values can be used
to construct the matrix $Z$ accurate to an arbitrary factor on the left-hand
side, viz. unitary matrix $U$ in polar expansion $Z=HU$. The matrix $Z$ is
shown to be expandable to a set of bispinors, for which the unitary matrix $U$
is responsible for the internal (gauge) degrees of freedom. Thus, a group of
gauge transformations depends only on the Riemannian space dimension,
signature, and the number field used. The constructed algorithm for mapping
tensors to bispinors admits extension to Riemannian spaces of a higher
dimension.
