“If you cut the snake’s head off, the body will die.’”

— Gregg Williams, to his Redskins’ defensive players, circa 2004-2007

One of the most illuminating moments in Gregg Williams’s NFL career of controlled mayhem happened when he was interviewing to become Joe Gibbs’s replacement as head coach of the Washington Redskins in January 2008. From Daniel Snyder’s Potomac mansion, he texted a reporter how great the all-day interview was going, intimating that he expected to be given the job perhaps that day by Snyder and his right-hand man, Vinny Cerrato.

Beyond showing remarkable hubris — and incredible ignorance about a head-coaching job he was never going to get — the anecdote said everything about Williams: He thought more of himself than others thought of him and he believed deeply that the force of his own personality could trump all the black marks and bad-character references against him.

As usual, Gregg Williams was wrong.

Though he was congenial to almost everyone he needed in the media, we used to derisively say in Ashburn that the third “G” in Williams’s first name stood for “genius,” and if you want to find out how great and smart the team’s defensive coordinator was, well, just ask him.

Today when NFL Commissioner Roger Goodell’s investigative team brought Williams and the New Orleans Saints up on bounty-hunting charges — the same pay-for-pain program he administered with the Redskins from 2004 to 2007, multiple players told The Post on Friday — the third “G” should stand for “goon.”

Or “gutter,” the proper place for warped coaches who push the envelope of an already violent game way too far.

Or “god complex,” which is what several former players said Williams had.

From 2009 to 2011, the Saints’ “program” paid $1,500 for a “knockout” hit and $1,000 if an opposing player was carted off the field, the NFL said. The system worked the same way in both New Orleans and Washington apparently, in which Williams would fine players $500 for being late to practices and meetings before redistributing it. He would contribute some of the money from his own pocket.

Now the St. Louis Rams’ defensive coordinator, Williams is frightened he could cost his new employers part of their future.

Williams apologized Friday for what he called a “pay for performance” program while he was with the Saints, calling it a “terrible mistake” and taking full responsibility for his role in a prepared statement.

Damage control through a public apology is not enough. Goodell needs to suspend Williams for a full season, if not more — take away his livelihood for a while the way he promoted taking other livelihoods away.

And the teams that employed him and knew of his bounty on many of the game’s stars — and knowingly supported it — need to lose draft picks and be reprimanded for terrible misjudgment.

Failure to punish sternly would tell all those players writhing on the ground in agony that they don’t matter, they’re expendable whenever an egomaniacal coordinator believes he’s bigger than the tenets of the game.

This was uncontrolled mayhem.

Before “No big deal” and “Bounties have been around forever” is echoed — in this musclehead subculture, it’s already too late for that — let’s be clear: Gregg Williams wasn’t awarding players for merely sacks and fumbles.

“He called it a ‘kill shot,’ meaning you got good money for taking another player out of the game who meant something to that team,” a Redskin who played for Williams said on condition of anonymity. “I never got money myself, but I know folks who did. Sean Taylor made more money than all of us, I can tell you that.”

As much as pro football wants to prevent concussions and long-term brain and body injuries, there is only so much medical care and compassion available when your job description involves physically emasculating another man.

But there is a difference between giving out a “Hit Stick,” awarded to the Redskin who made the biggest hit each week by current special teams coach Danny Smith, and Williams’s open bounties on big-name offensive stars.

For one, it’s banned by the agreement between the league and its players. And in the case of Saints Coach Sean Payton, who knew of the program, it represents a lack of institutional control, an almost renegade mentality that corrupts a sport already carting body after body off the field without bounties.

Maybe Williams becomes the first of many overzealous coaches to get caught in the NFL for this practice. Maybe the Browns letting Colt McCoy back on the field after a malicious James Harrison dropped him like a rag doll this past December is a worse crime against the game.

But another anecdote that tells you how much larger than the game Williams thought he was, how crass and out of line he has conducted himself in a job that promotes wild-eyed men routinely for their ugly side:

Williams never told Gibbs, the supposed man in charge, that he was planning to put 10 men on the field to honor Taylor, the missing man, on the first play of the first game after Taylor’s death. It’s not an offense worth firing someone over, but it showed a clear disregard for the authority of a Hall of Fame coach who gave Williams a job in Washington.

Gregg Williams has been acting above the rules for too long. Now that’s he found out, it’s time to send a real message in an NFL sick and tired of seeing its former players slur their words and hobble around on arthritic limbs:

Cut off the coach, and the career will die.

For Mike Wise’s previous columns, go to washingtonpost.com/wise.