  We use numerically exact diagonalization to calculate the spin-orbit and
phonon-induced triplet-singlet relaxation rate in a two-electron quantum dot
exposed to a tilted magnetic field. Our scheme includes a three-dimensional
description of the quantum dot, the Rashba and the linear and cubic Dresselhaus
spin-orbit coupling, the ellipticity of the quantum dot, and the full angular
description of the magnetic field. We are able to find reasonable agreement
with the experimental results of Meunier et al. [Phys. Rev. Lett. 98, 126601
(2007)] in terms of the singlet-triplet energy splitting and the spin
relaxation rate, respectively. We analyze in detail the effects of the
spin-orbit factors, magnetic-field angles, and the dimensionality, and discuss
the origins of the remaining deviations from the experimental data.
