This entry carries a name that merges two things the Golden Dawn kept apart, and the distinction is worth stating because most popular lists lose it.
The Air Dagger is the elemental weapon of Air and belongs to the Theoricus grade. It is small, made by the aspirant, painted yellow with purple lettering in the flashing pair for Air, and it is consecrated for the same class of use as the other three elemental implements: to represent and to direct one element within the practitioner's own work.
The Magic Sword, in the same system, is not an elemental weapon at all. It is an implement of force and defence, used in banishing and in the more assertive parts of ceremonial, and it is not tied to a single grade or element in the way the four are. The grimoire tradition behind the order describes a sword of its own, with instructions in the Key of Solomon for its making and inscription, and that sword is likewise a working tool rather than a symbol of Air.
Both carry the same broad association, with the discriminating intellect that separates one thing from another, which is why they collapse together so easily. Keeping them apart matters in practice because the two are used at different points and, in the order's own instructions, consecrated by different rites. Where a modern list offers a single sword standing for Air, it is following a simplification rather than the sources.
Sources The Golden Dawn: The Original Account of the Teachings, Rites, and Ceremonies of the Hermetic Order of the Golden Dawn, 6th editionIsrael Regardie with Grimoires: A History of Magic BooksOwen Davies