For the past few decades, the Islanders organization has floated further and further away from the traditions and the team that made up the greatest run in sports history. The dynasty of the four straight Stanley Cups, starting in 1980, and the players who made up those teams, were becoming forgotten by the younger generation of the fan base.

But on Monday, at the team’s sparkling new practice facility on Long Island, with whip-smart new owner Jon Ledecky leading the show, and with one more memorial to beloved coach Al Arbour, who passed away one year and one day ago, things are beginning to look stabilized.

Finally, it seems this is a franchise that isn’t just talking about honoring its history, but is doing it in as many ways as possible. And that starts with bringing back Islanders alumni and having them as engaged as possible, as seen in their teary eyes as they once again shared memories of their coach, who died last Aug. 28 and continues to set the standard for integrity in a sometimes wayward organization.

“When you look at a world of [salary] cap and free agency, and you look at the discussions people have about teams, having an engaged alumni base is a terrific, positive mark for people deciding where they’re going to play,” said Ledecky, who is becoming the face of this front office.

“There is tremendous heritage and tradition of the Islanders, and there is also the future.”

For now, that future is in Brooklyn, and Ledecky remained coy about the swirling rumors of a relocation after another three years, when the team’s contract with Barclays Center has an out clause.

“Last time I looked, I am not the predictor of the future,” Ledecky said in another measured comment that leans heavier on those running the building — namely Barclays CEO Brett Yormark — to bend to his demands.

That surely seems to be happening. Ledecky said Barclays had provided an improved ice surface, an improved fan experience and better facilities and transportation for players’ families. And there is also going to be a suite, next to the owners’ suite, for Islanders alumni to come to any game free of charge — one more way Ledecky is trying to “reinvigorate the alumni.”

“The current players being able to interact with the Islanders greats, they’re learning from each other,” Ledecky said. “I think you saw [at the memorial], with the speeches that were given, and you talk to the [current] players after, and they’re going, ‘This was really something, huh?’”

It really was.

After talks by Bobby Nystrom, Mike Bossy, Ken Morrow, Butch Goring, Denis Potvin and the show-stopper by “The Architect” himself, Bill Torrey, the connection to the history of the Islanders was unmistakable. Among those in the audience was captain John Tavares, who said, “It gives you the chills for sure.” Defenseman Johnny Boychuk was nearby, and he told Ledecky he learned a lot from the alumni “about what constitutes greatness.” Even Jason Chimera, a new free-agent signing, was there to pay his respects.

“I think as players today, part of the Islanders now is recognizing the standard and the tradition of the Islanders and what it means to the community,” Tavares said. “And a guy like Al Arbour, what he meant to the organization and the community here, you certainly want to carry that on and bring that excellence on and off the ice that he certainly brought.”

That excellence now hinges on the work of Tavares and those who surround him, chosen by general manager Garth Snow. In a very grown-up offseason, Snow let three homegrown talents and fan favorites walk out the door as pricey free agents — Kyle Okposo, Frans Nielsen and Matt Martin. He partially replaced them with the likes of Chimera, Andrew Ladd and old buddy P.A. Parenteau. That goes along with the hope that young players such as Matthew Barzal, Michael Dal Colle and possibly Josh Ho-Sang can show the ability to contribute once training camp commences in a month.

“We want to win,” Snow told The Post. “When you’re part of an organization that has the history that the New York Islanders have, you’re always going to remember those Stanley Cup teams, the dynasty that they had. It’s a model for not only us and what we should strive to be, but it’s a model for every NHL team, to try to have the success those great teams had.”

As Ledecky brings the past and the future closer together, he hopes that success follows. This is the closest the Islanders have been to that in some time.