  Classical random matrix ensembles with orthogonal symmetry have the property
that the joint distribution of every second eigenvalue is equal to that of a
classical random matrix ensemble with symplectic symmetry. These results are
shown to be the case $r=1$ of a family of inter-relations between eigenvalue
probability density functions for generalizations of the classical random
matrix ensembles referred to as $\beta$-ensembles. The inter-relations give
that the joint distribution of every $(r+1)$-st eigenvalue in certain
$\beta$-ensembles with $\beta = 2/(r+1)$ is equal to that of another
$\beta$-ensemble with $\beta = 2(r+1)$. The proof requires generalizing a
conditional probability density function due to Dixon and Anderson.
