  To fully extract cosmological information from nonlinear galaxy distribution
in redshift space, it is essential to include higher-order statistics beyond
the two-point correlation function. In this paper, we propose a new
decomposition formalism for computing the anisotropic bispectrum in redshift
space and for measuring it from galaxy samples. Our formalism uses tri-polar
spherical harmonic decomposition with zero total angular momentum to compress
the 3D modes distribution in the redshift-space bispectrum. This approach
preserves three fundamental properties of the Universe: statistical
homogeneity, isotropy, and parity-symmetry, allowing us to efficiently separate
the anisotropic signal induced by redshift-space distortions (RSDs) and the
Alcock-Paczy\'{n}ski (AP) effect from the isotropic bispectrum. The relevant
expansion coefficients in terms of the anisotropic signal are reduced to one
multipole index $L$, and the $L> 0$ modes are induced only by the RSD or AP
effects. Our formalism has two advantages: (1) we can make use of Fast Fourier
Transforms (FFTs) to measure the bispectrum; (2) it gives a simple expression
to correct for the survey geometry, i.e., the survey window function. As a
demonstration, we measure the decomposed bispectrum from the Baryon Oscillation
Spectroscopic Survey (BOSS) Data Release 12, and, for the first time, present a
$14\sigma$ detection of the anisotropic bispectrum in the $L=2$ mode.
