Washington DC - Far-right billionaire Elon Musk said Tuesday his critics needed "better dirty tricks" after he caused outrage by giving Nazi salutes at President Donald Trump's inauguration event.

The X, SpaceX, and Tesla chief appeared on stage at the Capital One Arena in Washington, where supporters of the newly inaugurated president had gathered for a rally.

After hanking the crowd for returning Trump to the White House, Musk tapped the left side of his chest with his right hand and then extended his arm with his palm open, repeating the gesture for the crowd seated behind him.

Claire Aubin, a historian who specializes in Nazism in the US, said Musk's gesture was a "sieg heil," or Nazi salute.

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"My professional opinion is that you're all right, you should believe your eyes," Aubin posted on X.

Ruth Ben-Ghiat, a historian of fascism, found the gesture "was a Nazi salute – and a very belligerent one too."

In response, Musk said on X that his opponents needed "better dirty tricks."

"The 'everyone is Hitler' attack is sooo tired," he added.

Musk has been vocal in his support for Germany's far-right AfD party, and has for years boosted right-wing and outright fascist accounts on his X platform.

Meanwhile, the Anti-Defamation League, an organization founded to combat antisemitism, defended his behavior, calling it "an awkward gesture in a moment of enthusiasm, not a Nazi salute."

It's not the first time the ADL has stood up for Musk, who previously threatened to sue the organization. Its CEO, Jonathan Greenblatt, has been denounced by staff for targeting critics of Israel while cozying up to powerful figures with a history of making or sharing anti-Jewish remarks.

Reports by Wired and Rolling Stone said far-right and neo-Nazi figures in the US were celebrating the move.