In the first nontrivial case, dimension $p = 3$ and sample size $N = 3$ or 4 (depending on whether or not the mean is known), it is proved that the classical level $\alpha$ normal test of independence of the first component from the others, based on the squared sample multiple correlation coefficient $R^2$, maximizes, among all level $\alpha$ tests, the minimum power on each of the usual contours where the $R^2$-test has constant power. A corollary is that the $R^2$-test is most stringent of level $\alpha$ in this case.
"Minimax Character of the $R^2$-Test in the Simplest Case." Ann. Math. Statist. 35 (4) 1475 - 1490, December, 1964. https://doi.org/10.1214/aoms/1177700374