  All the segments (or their continuations) of a billiard trajectory inside an
ellipsoid of $\Rset^n$ are tangent to n-1 quadrics of the pencil of confocal
quadrics determined by the ellipsoid. The quadrics associated to periodic
billiard trajectories verify certain algebraic conditions. Cayley found them in
the planar case. Dragovi\'{c} and Radnovi\'{c} generalized them to any
dimension. We rewrite the original matrix formulation of these generalized
Cayley conditions as a simpler polynomial one. We find several remarkable
algebraic relations between caustic parameters and ellipsoidal parameters that
give rise to nonsingular periodic trajectories.
