In the slice Hardy space over the unit ball of quaternions, we introduce the slice hyperbolic backward shift operator S with the decomposition process f=ea〈f,ea〉+Ba∗Saf,where e denotes the slice normalized Szegö kernel and B the slice Blaschke factor. Iterating the above decomposition process, a corresponding maximal selection principle gives rise to the slice adaptive Fourier decomposition. This leads to a adaptive slice Takenaka–Malmquist orthonormal system.