I have a question regarding the use of Basu's Lemma. Assume that $$X \sim N(\theta, \sigma^2).$$ In that case, we can use Basu's Lemma to show that $$\bar{X}$$ and $$S^2= \frac{1}{n-1}\sum(X_i-\bar{X})^2$$ are independent, if we show that $T:=\bar{X}$ is complete and sufficient (e.g. by the definition of exponential families) and that the distribution of $S^2$ does not depend on $\theta$ which is the case here. However, when $\sigma^2$ would be unknown as well, the distribution of $S^2$ would depend on $\theta$. Can we still use Basu's Lemma somehow to show independence in that case ? I was confused by the following statement in the book Statistical Inference by Casella and Berger on p.289:
