Then why did he publicly do a Nazi salute?
He did not do a Nazi salute because otherwise ADL would've been all over him. ADL came out saying he didn't do a Nazi salute.
The ADL's morally bankrupt calculus is that they value support for Israel more, so they covered for Musk as long as he supports Israel. Which sadly, makes the rest of us less safe.
And they were criticised for this by many other Jewish organisations.
The ADL is not the final arbitrator of what is a Nazi salute and what isn't.