The Canadiens received William Lagesson, a conditional second-round pick in the 2022 NHL Draft and a seventh-round pick in the 2024 NHL Draft in return for the defenseman. If the Oilers reach the Stanley Cup Final, the second-round pick in the 2022 draft will become a second-round pick in the 2023 NHL Draft.
Montreal will retain 50 percent of Kulak's salary for the remainder of the season.
"At the end of the day, we were looking for a guy that could be a top six defenseman]," Oilers general manager Ken Holland said." He played last year, 13 ... playoff games. At stretches this year, he's been in the first pair with [Jeff] Petry, he had matched up against first lines. He comes in and he can play anywhere from the top four to the third pair. So, I think he gives us versatility and he's mobile, has good size and we want to get the puck moving forward."
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Kulak is in the final season of a three-year, $5.55 million contract ($1.85 million average annual value) he signed with the Canadiens on May 25, 2019. He can become an unrestricted free agent after the season.
The 28-year-old has scored 13 points (three goals, 10 assists) in 56 games this season.
Selected by the Calgary Flames in the fourth round (No. 105) of the 2012 NHL Draft, Kulak has scored 56 points (13 goals, 43 assists) in 316 regular-season games for the Canadiens and Flames and four assists in 23 Stanley Cup Playoff games.
Lagesson has four assists in 30 games this season. Selected in the fourth round (No. 91) of the 2014 NHL Draft by the Oilers, the defenseman has six assists in 57 games.
The Oilers acquired forward Derick Brassard in a trade with the Philadelphia Flyers for a fourth-round pick in the 2023 NHL Draft.
NHL.com independent correspondent Derek Van Diest contributed to this report