Suppose I have a regression model: $$y_{i}=x_{1i}\beta_{1}+x_{1i}D_{i}\beta_{2}+\epsilon $$ where $\mathbb{E}\left[\epsilon_{i}|x_{i}\right]\neq0$ , and there is a problem of endogeneity. In the above, think of the dummy variable being equal to $1$ when $i$ is male. As such, this model allows for a differential effect of $x_{1i}$ on $y_{i}$, if the individual is a male or female. In order to obtain consistent estimates, suppose I have an instrument $z_{1i}$, which is both a relevant instrument for $x_{1i}$, and satisfies the exclusion restriction (such that $cov(z_{i},\epsilon_{i})=0).$ If there were no interaction term, it would be trivial to estimate the above using Two stage least squares, or simply just computing: $$ \hat{\beta}=\left(\boldsymbol{z'}x\right)^{-1}\left(\boldsymbol{z'y}\right) $$
However, the complication arises due to the interaction term. How would I go about obtaining estimates for both $\beta_{1}$ and $\beta_{2}$? One option is to construct another instrument $D_{i}Z_{1i}$. Would such a procedure be valid?