The partition

We first note that

$$[H_i(t'),\rho_{se}(t')]=H_i(t')\rho_{se}(t')-\rho_{se}(t')H_i(t')$$

which we use to write $\left[H_i(t),[H_i(t'),\rho_{se}(t')]\right]=$

$$\overbrace{H_i(t)H_i(t')\rho_{se}(t')}^{A}-\underbrace{H_i(t)\rho... ...erbrace{H_i(t')\rho_{se}(t')H_i(t)}^C+\underbrace{\rho_{se}(t')H_i(t')H_i(t)}_D$$

We will now proceed to treat the integration case by case.