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The Australian made three Premier League appearances for the Gunners and had been linked with a permanent move to the Emirates.
However, the 29-year-old will now return to LaLiga, subject to international clearance, having joined Brighton from Valencia back in 2017.
“Mat has been a great servant to the Albion over the past four years and on behalf of the club I’d like to place on record our gratitude for his efforts,” Seagulls boss Graham Potter told the club website.
“He’s a great guy, top professional and as someone who wants to be playing regularly at senior level he goes to Real Sociedad with our very best wishes.
“He’s been a pleasure to work with, he will be fondly remembered by everyone and always welcome back at the club.”