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In this paper, the authors study the problem of testing the hypothesis of
a block compound symmetry covariance matrix with two-level multivariate observa-
tions, taken for m variables over u sites or time points. Through the use of a suitable
block-diagonalization of the hypothesis matrix, it is possible to obtain a decomposi-
tion of the main hypothesis into two sub-hypotheses. Using this decomposition, it is
then possible to obtain the likelihood ratio test statistic as well as its exact moments in
a much simpler way. The exact distribution of the likelihood ratio test statistic is then
analyzed. Because this distribution is quite elaborate, yielding a non-manageable dis-
tribution function, a manageable but very precise near-exact distribution is developed.
Numerical studies conducted to evaluate the closeness between this near-exact distrib-
ution and the exact distribution show the very good performance of this approximation
even for very small sample sizes and the approach followed allows us to extend its
validity to situations where the population distributions are elliptically contoured. A
real-data example is presented and a simulation study is also conducted.