The All Blacks lost the test by a record margin - 47-26.
Williams was handed a four-week ban for a no arm tackle on Anthony Watson in the second Lions test in 2017.
A similar ban would probably rule Barrett out of the final two tests before the Rugby World Cup against Australia next week in Auckland and Tonga in Hamilton on September 7.
How our reporters saw it:
Patrick McKendry - Just a yellow: Barrett was off balance and connected only a glancing blow on Hooper's head area – it was difficult to tell where exactly - as the Wallabies skipper drove low from an attacking ruck. But unfortunately for Barrett and the All Blacks the result was the same; a man down, the odds all against them and a controversial defeat after positive beginnings.
Gregor Paul - Just a yellow: Poor old Barrett can consider his red card one of the least deserved in history. A yellow would have surely appeased Garces' need to do something dramatic and impose himself on the game and yellow would have at least given the contest what it needed to flourish as a two-way street.
Liam Napier - Just a yellow: Barrett may have deserved a yellow card at most for his shoulder charge just before half time which made contact with Michael Hooper's head. But slow down every pick and drive during any match and you will find similar instances in such close quarters at knee height. Do we really want to be sending players off willy-nilly and near ruining every contest? That is where this sort of officiating will end up taking us.
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