  We study a general one-mode non-Hermitian quadratic Hamiltonian that does not
exhibit $\mathcal{PT}$-symmetry. By means of an algebraic method we determine
the conditions for the existence of real eigenvalues as well as the location of
the exceptional points. We also put forward an algebraic alternative to the
generalized Bogoliubov transformation that enables one to convert the quadratic
operator into a simpler form in terms of the original creation and annihilation
operators. We carry out a similar analysis of a two-mode oscillator that
consists of two identical one-mode oscillators coupled by a quadratic term.
