Takagi’s decomposition of a symmetric unitary matrix as a finite algorithm

Takagi’s decomposition is an analog (for complex symmetric matrices and for unitary similarities replaced by unitary congruences) of the eigenvalue decomposition of Hermitian matrices. It is shown that, if a complex matrix is not only symmetric but is also unitary, then its Takagi decomposition can be found by quadratic radicals, that is, by means of a finite algorithm that involves arithmetic operations and quadratic radicals. A similar fact is valid for the eigenvalue decomposition of reflections, which are Hermitian unitary matrices.