The elemental archangels are the youngest of the three sets in this collection and the only one with a single identifiable point of origin. They come from the quarters of the Lesser Banishing Ritual of the Pentagram as the Hermetic Order of the Golden Dawn taught it in the late nineteenth century, where the operator names four angels around the circle: Raphael before, Gabriel behind, Michael on the right hand and Auriel on the left.
Facing east, as the ritual assumes, that arrangement puts Raphael in the east with Air, Michael in the south with Fire, Gabriel in the west with Water and Uriel in the north with Earth. The whole elemental set follows from the operator's orientation.
Fire takes Michael for reasons the tradition never needed to argue. The scriptural Michael carries a sword, leads armies and casts out the dragon; the Fire of the Western elemental scheme is will, ardour and the initiating force, and its ritual weapon is the wand or the sword depending on the school.
The attribution is also unusually well behaved. Michael is solar in the planetary table and fiery here, and Fire is the solar element, so two of the three systems agree about him. Only the sephirotic placement in Hod, the sphere of Mercury, breaks the pattern.
Uniformity across modern practice is the mark of a young system with one source. The planetary and sephirotic sets, which are older and travelled by several routes, are nothing like as stable.