Extensions of higher local fields determined by certain matrix equations introduced by E. Inaba are studied. It is proved that any extension decomposable into a tower of Artin–Schreier extensions can be embedded into an Inaba extension that is a composite of the given extension and another Inaba extension. Furthermore, any p-extension with elementary Abelian Galois group can be embedded into an extension with the Galois group isomorphic to a group of unipotent matrices over a field with p elements. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.