Alexandrian Wicca is the largest of the Gardnerian offshoots, and its founding story is a documented falsehood, which is worth stating plainly, because the tradition itself has largely stopped repeating it.
Alex Sanders said he had been initiated as a child by his Welsh grandmother, Mary Bibby, having caught her at ritual. The account fails on dates: Bibby died in 1907, some nineteen years before Sanders was born in 1926. Research into his actual entry into the Craft points to a Gardnerian initiation in the early 1960s, from a priestess of that line. Alexandrian Wicca is therefore, in origin, a Gardnerian tradition that acquired a name of its own: and acquired it retroactively, around 1970, when Stewart Farrar was writing about the coven he had joined.
The name is usually explained as honouring both Sanders and the library of Alexandria, and the second half of that is a fair description of the tradition's temperament. Where Gardner assembled from folk practice and Freemasonry, Sanders drew heavily on ceremonial magic and the Qabalah, and Alexandrian ritual carries considerably more of that apparatus (angelic names, Enochian material, the Tree of Life), than its parent does.
Sanders courted publicity on a scale Gardner never attempted, styled himself King of the Witches, and was regarded by much of the Craft as an embarrassment. He also, in the process, taught a great many people, and the tradition has outlasted the persona comfortably, which is a better outcome than the founding story would lead one to expect.