The sums of squares and cross products of groups of random variables drawn from a multivariate normal population whose covariance matrix has a certain pattern indicating intra-class correlations are shown to follow the Wishart distribution. It follows immediately from this result that the usual distributions of the simple, partial, and multiple correlation coefficients obtain, although with non-centrality parameters that reflect the effect of the intra-class correlations.
"On the Distribution of Sums of Squares and Cross Products of Normal Variates in the Presence of Intra-class Correlation." Ann. Math. Statist. 33 (4) 1461 - 1463, December, 1962. https://doi.org/10.1214/aoms/1177704379