Consider three random variables $X_1,X_2,X_3$ and three dummy variables $D_1,D_2,D_3$ such that $D_1+D_2+D_3=1$.
Assume that $$ E(X_j| D_1,D_2,D_3)=0 \quad \text{ for each $j=1,2,3$} $$ Claim: $E(\sum_{j=1}^3 D_j X_j| D_1,D_2,D_3)=0$
Proof of claim: We have that $$ E(\sum_{j=1}^3 D_j X_j| D_1=1,D_2=0,D_3=0)=E(X_1|D_1=1,D_2=0,D_3=0)=0\\ E(\sum_{j=1}^3 D_j X_j| D_1=0,D_2=1,D_3=0)=E(X_2|D_1=0,D_2=1,D_3=0)=0\\ E(\sum_{j=1}^3 D_j X_j| D_1=0,D_2=0,D_3=1)=E(X_3|D_1=0,D_2=0,D_3=1)=0\\ $$ Hence, $E(\sum_{j=1}^3 D_j X_j| D_1,D_2,D_3)=0$.
Corollary: $E(D_k \sum_{j=1}^3 D_j X_j )=0$ for each $k=1,2,3$.
Proof of corollary: Let $Z\equiv \sum_{j=1}^3 D_j X_j$. By the above claim, $$ E(Z| D_1,D_2,D_3)=0. $$ Therefore, $E(Z D_k)=0$ for each $k=1,2,3$.
Question: I find the corollary result counterintuitive. In fact, it seems to claim that, e.g., the random variables $D_2$ and $\sum_{j=1}^3 D_j X_j$ are uncorrelated. However, $\sum_{j=1}^3 D_j X_j$ depends on $D_2$. Therefore, how can they be uncorrelated?