  Spin relaxation of a single electron in a weakly coupled double quantum dot
is calculated numerically. The phonon assisted spin flip is allowed by the
presence of the linear and cubic spin-orbit couplings and nuclear spins. The
rate is calculated as a function of the interdot coupling, the magnetic field
strength and orientation, and the dot bias. In an in-plane magnetic field, the
rate is strongly anisotropic with respect to the magnetic field orientation,
due to the anisotropy of the spin-orbit interactions. The nuclear spin
influence is negligible. In an out-of-plane field, the nuclear spins play a
more important role due selection rules imposed on the spin-orbit couplings.
Our theory shows a very good agreement with data measured in [Srinivasa, et
al., PRL 110, 196803 (2013)], allowing us to extract information on the linear
spin-orbit interactions strengths in that experiment. We estimate that they
correspond to spin-orbit lengths of about 5-15 $\mu$m.
