Using Stein's  generalisation of a method of Lehmann and A. Birnbaum (A. Birnbaum ), we show that two of the tests commonly used in multivariate analysis of variance are admissible.
"On the Admissibility of Some Tests of Manova." Ann. Math. Statist. 35 (2) 789 - 794, June, 1964. https://doi.org/10.1214/aoms/1177703578