Extending the non-relativistic string AdS coset

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.