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.