Elon Musk, during the party at the Capital One Arena for Donald Trump's inauguration, concluded his speech with an arm extended towards the audience, repeating the gesture twice.
The interpretations derived from the analysis of the repeated gesture are controversial.

I believe that, given that Musk is autistic, more precisely suffering from Asperger's syndrome, it is possible that the gesture was involuntary. 
It is important to distinguish between behaviors associated with the syndrome and actions that could be interpreted as having a political or social connotation.
People with Asperger's syndrome may have difficulty understanding social context; some gestures, in fact, could be interpreted out of context or used without full awareness of their social or historical meaning. This syndrome can also involve repetitive or stereotyped attitudes, performed without necessarily understanding their symbolic value.

In the case of Elon Musk, a gesture like the "supposed Roman salute" could be an action lacking awareness of historical context, a cultural misunderstanding (which I personally think is unlikely), or a misinterpretation by observers. 
In fact, Musk continued his speech with the phrase "My heart goes out to you", which literally means "My heart is with you". It is therefore plausible that the gesture of the hand on the chest and then extended towards the audience was made in good faith.

This does not mean, however, that it is important that such a relevant figure should be aware of what he shows to the public and to the people who follow him every day.

Elon Musk's presumed Roman Salute