A problem in ANOVA, I think the result will be useful but I do not know how to prove it.

Question

Consider an ANOVA model: $$y_{ijl}=\mu+\alpha_{i}+\beta_{j}+\epsilon_{ijl}$$ where $i=1,2\dots I, j=1,2\dots J, l=1,2\dots n_{ij}, \epsilon_{ijl}\sim N(0,\sigma^2) $i.i.d, we also have the constrains to ensure identifiability: $$\sum_{i=1}^{I}\alpha_i=0, \sum_{j=1}^J\beta_j=0$$ Show that the estimations for $\alpha_i$ and $\beta_j$, which are denoted by $\hat{\alpha}_i, \hat{\beta}_j$, are uncorrelated if and only if the following holds: $$n_{ij}=\frac{n_{i.}n_{.j}}{n_{..}}$$ where: $$n_{i.}=\sum_{j=1}^Jn_{ij}, n_{.j}=\sum_{i=1}^In_{ij}, n_{..}=\sum_{i,j}n_{ij}$$ (I think the result is consice and beautiful, but I cannot figure it out. I guess there must be some tricks, but I don't know where they are!)