To answer this question, it seemed logical to begin with how Islamists approach their archenemy: Israel and the Jews. Thinking this may be difficult to prove — it is one thing to hate your enemy, another to project unconsciously your worst traits onto him — I expected this might require some research. I went to MEMRI and, lo and behold, came across three back-to-back examples of projection against Israel and the Jews. Consider the following excerpts, especially the italicized portions:
On September 7, Egyptian cleric Abdallah Samak made the following remarks on Al Rahma TV: 'The Jews are known for their merciless, murderous, and bloodthirsty nature… The number one characteristic of the Jews – which appears in the Bible – is that they are always prepared for combat. They believe that it is their fate and destiny to be in a state of perpetual war. This is not what we want. We are seekers of peace and security. We seek to spread love. But we are dealing with a people, a society, that believes that its destiny is linked to war. The number one characteristic of the Jews is that they are a people that believes that its destiny is linked to war. They cannot live without war. They can only live if they attack others. They can only live through annihilation, revenge, and mercilessness.'
The notion of 'perpetual war,' in fact, is straight out of Muslim doctrine and history — best recognized by the word 'jihad'— and has no corollary in Judaism or any other religion. Even temporary truces are permissible only when Muslims are weak and incapable of going on the offensive: according to sharia, once Muslims are strong enough and have proper leadership (e.g., a caliph), they are obligated to expand the realm of Islam through offensive jihad until, in the words of the Koran, 'all religion belongs to Allah' (8:39). History unequivocally attests to this approach. Moreover, while the Old Testament certainly contains many allusions to violence, these are of a historical, as opposed to doctrinal, nature. Conversely, Koranic verses dealing with violence have been codified in sharia law, and thus have a juridical and perpetual quality (note the word 'until' in the most violent passages of the Koran, e.g., 9:5 and 9:29; see here for more on the differences between Judeo-Christian and Muslim violence). Finally, by quickly adding that Muslims 'do not want' perpetual war, but instead seek 'to spread love,' Samak belies the fact that Muslims naturally came to his mind immediately after he evoked 'perpetual war,' evincing a rather telling train of thought.
On October 8, Palestinian journalist Khaled Amayreh made the following remarks on Press TV: 'The Israelis are trying to divert the attention of the world from the Nazi-like crimes committed by Israel in the Gaza Strip and the West Bank, by inventing an imaginary threat called Islamism. The people who are trying to take over the world, including superpowers... you know... are Zionism, not Muslims. … The only people who are taking over Europe and the United States—they do that by lying and by mendacity — are the Zionist Jews. I want to just say something about the land... [The notion that God promised the land of Israel to the Jews attempts] to legitimize land theft in the name of God, just like those colonizers in North America, who killed millions of Native Americans, and they called it 'Manifest Destiny,' and they celebrated genocide on a day called Thanksgiving. So this man [co-debater] is trying to justify land theft, ethnic cleansing, and genocide by attributing all of this to the Almighty.'
Again, aspects that are part and parcel of Islam are projected onto the Jews. After bemoaning that the Jews have 'invented an imaginary threat called Islamism,' he himself evokes fairy tales of Jews taking over the world. Next, he ascribes 'lying and mendacity' to the Jews when, in fact, deception is a well codified aspect of sharia law, and permeates Islamist society. Finally, he accuses Jews and Americans of appealing to God 'to justify land theft, ethnic cleansing, and genocide,' when, truth be told, the story of Islam has been a story of conquests, land theft, and ethnic cleansing — all in the name of God. The overwhelming majority of what is today called the 'Muslim world' was taken from non-Muslims by violence and bloodshed. Even the Arab-Israeli conflict is a byproduct of the fact that Muslims took, by conquest (circa 638), a piece of land that, centuries earlier, belonged to the Jews. As for ethnic cleansing, history has forgotten the many peoples and languages that were either cleansed or absorbed by Islam. Modern examples of ethnic cleansing include Turkey's genocide of Christian Armenians and Khartoum's current 'cleansing' of Sudan's infidels (polytheists and Christians).
On October 10, Egyptian cleric Galal Al-Khatib, while comparing the Shi'as to the Jews, made the following comments on Al-Rahma TV: 'Jews accuse all their enemies of being infidels… The Jews believe that all non-Jews will end up in the Hellfire for all eternity… Both the Jews and the Shiites sanction the killing of those who disagree with them. Like the Jews, the Shiites employ treachery and deception to kill those who disagree with them. They use the same methods to get rid of their opponents. The Jews allow the plundering of their opponents' property…'
First, few religions are as keen on dividing the world between believers and infidels as Islam is, which divides the world into the Abode of Islam (where sharia is enforced) and the Abode of War (where it is not), holding that the armies of the former must wage war upon the latter — as we have seen, whenever they can. Moreover, through the doctrine of 'loyalty and enmity,' Muslims are commanded to disassociate themselves from non-Muslims. As for the Jews 'employ[ing] treachery and deception to kill those who disagree with them,' as mentioned, this is straight out of Muslim doctrine: Muhammad himself ordered the assassination of several poets for simply offending him; more to the point, he permitted the assassins to lie to their victims -- in order to win their trust and get close enough to them to murder them. Finally, Islam unequivocally legitimizes plundering the infidel — a longtime source of motivation for the soldiers of Islam.
According to one academic article devoted to projection and violence, 'Projection allows the killer to project his (unacceptable to him) desire to kill (torture, rape, steal, dominate, etc.) onto some target group or person. This demonizes his target, making it even more acceptable to kill.' Little wonder, then, that Israel and the Jews — the number one targets of all Islamists — are daily being portrayed by their antagonists as no better than Islamists themselves." (source)