  The algebraic method enables one to study the properties of the spectrum of a
quadratic Hamiltonian through the mathematical properties of a matrix
representation called regular or adjoint. This matrix exhibits exceptional
points where it becomes defective and can be written in canonical Jordan form.
It is shown that any quadratic function of $K$ coordinates and $K$ momenta
leads to a $2K$ differential equation for those dynamical variables. We
illustrate all these features of the algebraic method by means of the
Pais-Uhlenbeck oscillator and its PT-variant. We also consider a trivial
quantization of the fourth-order differential equation for the
quantum-mechanical dynamical variables.
