  We show how the one-mode pseudo-bosonic ladder operators provide concrete
examples of nilpotent Lie algebras of dimension five. It is the first time that
an algebraic-geometric structure of this kind is observed in the context of
pseudo-bosonic operators. Indeed we don't find the well known Heisenberg
algebras, which are involved in several quantum dynamical systems, but
different Lie algebras which may be decomposed in the sum of two abelian Lie
algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie
algebras) for this scope and find that it describes very well the behaviour of
pseudo-bosonic operators in many quantum models.
