Classically, formal orthogonal polynomials are studied with respect
to a linear functional, which gives rise to a moment matrix with a Hankel
in most situations, the moment matrix is supposed to be strongly regular.
This implies a number of algebraic
properties which are well known, like for example the existence of a
three-term recurrence relation (characterised by a tridiagonal Jacobi
matrix), Pad\'e approximation properties etc.
In this note we shall investigate how these formal algebraic properties
generalize for moment matrices with no special structure.
Subsequently, we shall look especially at the case of a moment matrix with an
indefinite Hankel structure
and with a nonsymmetric indefinite Toeplitz structure.