Mutation of signed valued quivers and presentations of simple complex Lie algebras

We define a mutation procedure for mutation Dynkin signed valued quivers which generalises Fomin-Zelevinsky mutation of skew-symmetrizable matrices appearing in the theory of cluster algebras. On forgetting the signs, this reduces to Fomin-Zelevinsky mutation. Each mutation Dynkin signed valued quiver gives a realization of the root system which is related to work of Parsons on companion bases, and it gives presentations of Lie algebras, following work of Barot-Rivera and P\'erez-Rivera.