Air is the element with the least stable set of attributions in the Western working traditions, and the instability is instructive rather than accidental.
In astrology air is a triplicity: Gemini, Libra and Aquarius. That grouping is settled, it is in Ptolemy, and no school disputes it. Everything downstream of it is less settled.
In the tarot, air is conventionally the suit of Swords and fire the suit of Wands, which is the Golden Dawn assignment and the one most modern decks follow. A substantial older current reverses them, giving the sword to fire as the forged and heated thing and the wand or rod to air as the breath-driven staff, and this is not a mistake in either direction: it is two schools reading the same objects by different logics. Anyone comparing two systems of correspondence should establish which convention each is using before concluding that one of them has made an error.
In the Golden Dawn system air is placed in the east, given the dagger as its working implement and the archangel Raphael as its regent. The eastern placement is the one attribution here that reaches back further, since dawn, breath and the beginning of the day form an old cluster.
The element's stable core, across every school that disagrees about the rest, is mind, speech and relation: air is what passes between things rather than what any one thing is made of. Aristotle gives it hot and wet, the qualities of expansion and of taking the shape of what holds it.
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 TetrabiblosClaudius Ptolemy