Using Roy's union-intersection principle, a unified treatment is developed for the construction of multivariate tests. These include Wilks' determinantal criteria, Hotelling-Lawley trace criterion, and Roy's largest characteristic root criterion. The key lies in the extension of an index set from vectors to matrices plus the use of elementary symmetric functions of characteristic roots to test component hypotheses.
"A New Class of Multivariate Tests Based on the Union-Intersection Principle." Ann. Statist. 9 (4) 792 - 802, July, 1981. https://doi.org/10.1214/aos/1176345519