The first four moments of the sum of $s$ non-null latent roots of a matrix occurring in multivariate analysis are studied. In particular, the first four moments of the sum of six roots are found, and are used to compare the upper percentage points obtained directly from the moment ratios with those from Pillai's approximate distribution.
"On the Moments of the Trace of a Matrix and Approximations to its Distribution." Ann. Math. Statist. 30 (4) 1135 - 1140, December, 1959. https://doi.org/10.1214/aoms/1177706097