  Lie algebra expansion is a technique to generate new Lie algebras from a
given one. In this paper, we apply the method of Lie algebra expansion to
superstring $\sigma$-models with a $\mathbb{Z}_4$ coset target space. By
applying the Lie algebra expansion to the isometry algebra, we obtain different
$\sigma$-models, where the number of dynamical fields can change. We reproduce
and extend in a systematic way actions of some known string regimes (flat
space, BMN and non-relativistic in AdS$_5 \times$S$^5$). We define a criterion
for the algebra truncation such that the equations of motion of the expanded
action of the new $\sigma$-model are equivalent to the vanishing curvature
condition of the Lax connection obtained by expanding the Lax connection of the
initial model.
