  We theoretically study the $(1,1)$ triplet to $(0,2)$ singlet relaxation rate
in a lateral gate-defined double quantum dot tuned to the regime of Pauli spin
blockade. We present a detailed derivation of the effective phonon density of
states for this specific charge transition, keeping track of the contribution
from piezoelectric as well as deformation potential electron-phonon coupling.
We further investigate two different spin-mixing mechanisms which can couple
the triplet and singlet states: a magnetic field gradient over the double dot
(relevant at low external magnetic field) and spin-orbit interaction (relevant
at high field), and we also indicate how the two processes could interfere at
intermediate magnetic field. Finally, we show how to combine all results and
evaluate the relaxation rate for realistic system parameters.
