This paper treats a generalization of the classical MANOVA testing problem. The problem is reduced via invariance considerations and a new test statistic is proposed. This new test is shown to be a unique locally best invariant test and locally minimax.
"The General MANOVA Problem." Ann. Statist. 6 (1) 200 - 214, January, 1978. https://doi.org/10.1214/aos/1176344079