Pillai and Mijares  gave the exact expressions for the first four moments of the sum of $s$ non-zero roots of a matrix occurring in multivariate normal analysis as studied independently by R. A. Fisher , P. L. Hsu  and S. N. Roy . In this paper some properties of completely homogeneous symmetric functions and certain determinantal results (Section 2) are used to give an inverse derivation of those moments (Section 4). The method is further extended to the moments in general of elementary symmetric functions (e.s.f.) of the roots of a matrix in multivariate analysis (Section 6) through the use of certain properties of compound matrices (Section 5).
"The Moments of Elementary Symmetric Functions of the Roots of a Matrix in Multivariate Analysis." Ann. Math. Statist. 32 (4) 1152 - 1160, December, 1961. https://doi.org/10.1214/aoms/1177704853