  We present a detailed calculation of our previous short paper [M. Shiraishi,
D. Nitta, S. Yokoyama, K. Ichiki, and K. Takahashi, Phys. Rev. D 82, 121302
(2010).] in which we have investigated a constraint on the magnetic field
strength through comic microwave background temperature bispectrum of vector
modes induced from primordial magnetic fields. By taking into account full
angular dependence of the bispectrum with spin spherical harmonics and Wigner
symbols, we explicitly show that the cosmic microwave background bispectrum
induced from the statistical-isotropic primordial vector fluctuations can be
also described as an angle-averaged form in the rotationally invariant way. We
also study the cases with different spectral indices of the power spectrum of
the primordial magnetic fields.
