The lifting formula under conditions (i)--(iii)

Let $\mathfrak A$ be an associative algebra with a trace, and let $D_1,\dots,D_i$ be trace-preserving associative derivations satisfying conditions (i)--(iii): each $D_j$ is trace-preserving, the lifting problem is considered for the monomial $D_1\wedge\dots\wedge D_i$, each commutator $[D_a,D_b]$ is an inner derivation $\operatorname{ad}Q_{ab}$ with $Q_{ab}\in\mathfrak A$, and $\operatorname{Alt}_{a,b,c}D_c(Q_{ab})=0$. The lifting conjecture. Under these conditions, the lifting problem can be solved for $\alpha^*_{D_1,\dots,D_i}$. The source presents this as a main-conjecture formulation of the construction, while later results provide cocycle formulas in the stated setting.