After Musk's recent behaviour, there has been much talk about the 'Roman salute' and its real origins. It seems to originate in the famous statute of the first Emperor, Augustus, although the arm was changed several times, due to it dropping off, and originally his hand had held a spear, so it was not a salute at all. https://images.app.goo.gl/mHqPkhEwXeXCH2VK7
Classical images became very popular during the Enlightenment and were then adopted by the French Revolutionaries and Napoleon, notably the office of Consul and the eagle symbols. The salute however is the work of David, who included it in his painting 'The Oath of the Horatii'
Yes, I remember David's painting. I have read that the salute (called the "Hitler Gruss" in German) was first adopted by the Italian fascists under Mussolini and then by the German Nazis. The Italian facsists may have adopted the salute from the imperial Roman salute.