Geodesic
Decomposition of the Multivariate Beta Distribution with Applications
Canadian mathematical bulletin, Tome 15 (1972) no. 2, pp. 225-228

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Let L be a positive definite symmetric matrix having a noncentral multivariate beta density of an arbitrary rank, see, e.g. Hayakawa ([2, p. 12, Equation 38]). Then an explicit procedure is given for decomposing the density of L in terms of densities of independent beta variates. 
Kabe, D. G.; Gupta, R. P. Decomposition of the Multivariate Beta Distribution with Applications. Canadian mathematical bulletin, Tome 15 (1972) no. 2, pp. 225-228. doi: 10.4153/CMB-1972-041-5
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     title = {Decomposition of the {Multivariate} {Beta} {Distribution} with {Applications}},
     journal = {Canadian mathematical bulletin},
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     year = {1972},
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     number = {2},
     doi = {10.4153/CMB-1972-041-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1972-041-5/}
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