  Inspired by Lie algebra expansion, we consider an extension of the algebra
introduced in arXiv:2203.07386 for the non-relativistic string coset action in
AdS$_5\times$S$^5$. We show that the extended algebra admits a non-degenerate
inner product which is adjoint invariant under the full extended algebra.
Furthermore, we provide a finite-dimensional representation of the extended
algebra.
