Within-Cluster Correlation
Note that $u_i$ is a random residue. In the linear regression model, we assume the independence of the random residue (error term). We have two slides above for case i.i.d.: $\epsilon_i \sim (0,\sigma^2)$, thus $\forall_{i\neq j}E[\epsilon_i\epsilon_j|X]=0$ and $E[\epsilon_i\epsilon_i|X]=\sigma^2$. Later (slide 10) we assume i.n.i.d. and $\forall_{g\neq g'}E[\epsilon_{ig}\epsilon_{jg'}]=0$.

Slide 12 shows the example matrix $\Omega=diag(\Sigma_g)$ that dispels doubts.