The triple differential cross section of the ${}^{208}{\mathrm{Pb}(}^{6}\mathrm{Li},\phantom{\rule{-0.16em}{0ex}}{\ensuremath{\alpha}d)}^{208}\mathrm{Pb}$ quasielastic breakup is calculated at a collision energy of 156 MeV and a scattering angle range of ${2}^{\ensuremath{\circ}}--{6}^{\ensuremath{\circ}}$. We fit the parameters of the Woods-Saxon potential using the experimental $\ensuremath{\alpha}\ensuremath{-}d$ phase shifts for different states to describe the relative motion of the $\ensuremath{\alpha}$ particle and deuteron. To check the validity of the two particle approach for the $\ensuremath{\alpha}\ensuremath{-}d$ system, we apply a potential model to describe the ${}^{2}\mathrm{H}{(\ensuremath{\alpha},\phantom{\rule{-0.16em}{0ex}}\ensuremath{\gamma})}^{6}\mathrm{Li}$ radiative capture. We calculate the Coulomb breakup using the semiclassical method while an estimation of the nuclear breakup is made on the basis of the diffraction theory. A comparison of our calculation with the experimental data of Kiener et al. [Phys. Rev. C 44, 2195 (1991)] gives evidence for the dominance of the Coulomb dissociation mechanism and the contribution of nuclear distortion, but is essentially smaller than the value reported by Hammache et al. [Phys. Rev. C 82, 065803 (2010)]. The results of our calculation for the triple cross sections (contributed by the Coulomb and nuclear mechanisms) of the ${}^{6}\mathrm{Li}$ breakup hint toward a forward-backward asymmetry in the relative direction of the $\ensuremath{\alpha}$ particle and deuteron emission, especially at smaller scattering angles, in the ${}^{6}\mathrm{Li}$ center-of-mass (c.m.) system.