  The wonderful formulas by I.Dumitriu and A.Edelman rewrite $\beta$-ensemble,
with eigenvalue integrals containing Vandermonde factors in the power $2\beta$,
through integrals over tridiagonal matrices, where $\beta$-dependent are the
powers of individual matrix elements, not their differences. These potentially
useful formulas are usually deduced from rather complicated and non-transparent
combinatorics and are not as widely known as they deserve. We explain that the
superintegrability property, i.e. a simple expression of the Gaussian averages
of arbitrary Jack polynomials through the same Jack polynomials, is immediately
consistent with this tridiagonal representation, which may serve as a clue to
its simple and transparent interpretation. For a formal non-perturbative proof,
we use the Virasoro constraints, which themselves acquire an interesting
structure in the tridiagonal realization. We also attract attention to the
surprising spontaneous breakdown of discrete invariance by the tridiagonal
measure, which may signal a new interesting anomaly at the elementary level of
the basic eigenvalue matrix model.
