Laser wakefield acceleration is paving the way for the next generation of electron accelerators, for their own sake and as radiation sources. A controllable dual-wake injection scheme is put forward here to generate an ultrashort triplet electron bunch with high brightness and high polarization, employing a radially polarized laser as a driver. We find that the dual wakes can be driven by both transverse and longitudinal components of the laser field in the quasiblowout regime, sustaining the laser-modulated wakefield which facilitates the subcycle and transversely split injection of the triplet bunch. Polarization of the triplet bunch can be highly preserved due to the laser-assisted collective spin precession and the noncanceled transverse spins. In our three-dimensional particle-in-cell simulations, the triplet electron bunch, with duration about 500 as, six-dimensional brightness exceeding 10^{14} A/m^{2}/0.1% and polarization over 80%, can be generated using a few-terawatt laser. Such an electron bunch could play an essential role in many applications, such as ultrafast imaging, nuclear structure and high-energy physics studies, and the operation of coherent radiation sources.