The authors derive the LRT statistic for the test of equality of mean vectors when the covariance matrix has what is called a double exchangeable structure. A second expression for this statistic, based on determinants of Wishart matrices with a block-diagonal parameter matrix, allowed for the expression of the distribution of this statistic as that of a product of independent Beta random variables. Moreover, the split of the LRT statistic into three independent components, induced by this second representation, will then allow for the expression of the exact distribution of the LRT statistic in a very manageable finite closed form for most cases and the obtention of very sharp near-exact distributions for the other cases. Numerical studies show that, as expected, due to the way they are built, these near-exact distributions are indeed asymptotic not only for increasing sample sizes but also for increasing values of all other parameters in the distribution, besides lying very close to the exact distribution even for extremely small samples.