I thought you might find interesting. From The New Republic:
To understand the emergence of Trump-supporting Straussianism, it’s important to realize that this group is very different than the Straussians who were influential during the Bush administration. After Leo Strauss died in 1973, his followers divided into two factions, creating the infamous “Crisis of the Strauss Divided.” And the best way to understand the divide between West Coast and East Coast Straussians is through the quarrel between Harry Jaffa and Allan Bloom, who were the respective heads of the rival schools.
Strauss encouraged his students to form tight relationships since frank and intimate conversation among friends was the heart of philosophy. Jaffa and Bloom were very close in the 1950s and early 1960s. In 1964 they co-wrote a book, Shakespeare’s Politics, dedicated “to Leo Strauss our teacher.” But over time Jaffa became involved in grassroots activism in the Republican Party, authoring the famous lines that Barry Goldwater uttered in 1964, “Extremism in the defense of liberty is no vice, and moderation in the pursuit of justice is no virtue.” As he became involved in right-wing activism, Jaffa gravitated towards social conservatism, praising the religious right, appearing on Pat Robertson’s show, and emerging as vocal homophobe (he argued in 1990 that “sodomy is, in the decisive respect, as morally offensive as incest and rape”). This put him in collision with his former friend Bloom, who was a closeted gay man. In a nasty review of Bloom’s Closing of the American Mind (1987) published in the Straussian journal Interpretation, Jaffa wrote that through AIDS “God and nature have exacted terrible retribution” on gays. Bloom died a few years after Jaffa wrote those vile words. Some friends, notably the novelist Saul Bellow, claimed that Bloom had died of AIDS, although this is disputed. What is undeniable is that Jaffa put a homophobic jab in his review with the intent to hurt his former friend.
The disputes between Jaffa’s West Coast Straussianism and Bloom’s East Coast Straussianism can be discussed along philosophic lines: Is America, as Jaffa believes, grounded in ancient philosophy or was the American founding, as Bloom would have it, built on the low but solid ground of early modern philosophers like Hobbes and Locke? Does the survival of America depend on the virtue of the people, as West Coast Straussians believe, or in the maintenance of constitutional norms, as East Coast Straussians believe? But the dispute can also more easily be understood in terms of the familiar social divide in the Republican Party. West Coast Straussians are the grassroots activists, grounded in social conservatism and ultra-nationalist in foreign policy. Sociologically, East Coast Straussians are more aligned with the party elite, and tend to be found in Washington think tanks and serving as career bureaucrats.
Another way to frame the divide is on the issue of regime change. Strauss, like Plato, was fascinated by the founding of regimes, and his students clearly believe that the key to politics is to have power at the moment of creation. For the East Coast Straussians, regime change is a matter of foreign policy, as witness the failed attempt to democratize the Middle East by force under Bush. For the West Coast Straussians—perhaps shaped by Jaffa’s Crisis of the House Divided, a seminal and brilliant work on Abraham Lincoln as a revolutionary thinker—regime change begins at home.
In a 1959 critique of the book in National Review, Willmoore Kendall prophetically argued that Jaffa’s celebration of Lincoln could give license to celebrations of new Caesars who claim to be avatars of the popular will. Readers of Jaffa’s book, Kendall warned, needed to be wary...
...lest Jaffa launch them, and with them the nation, upon a political future the very thought of which is hair-raising: a future made up of an endless series of Abraham Lincolns, each persuaded that he is superior in wisdom and virtue to the Fathers, each prepared to insist that those who oppose this or that new application of the equality standard are denying the possibility of self-government, each ultimately willing to plunge America into civil war rather than concede his point.
Kendall was wrong on one point. He feared that an America too beholden to the ideal of equality would see a rise in political extremism. But with West Coast Straussians supporting Trump, we see that Jaffa’s license to political extremism can be used just as easily by those who oppose equality—and Kendall’s warning of a new Caesarism has been fulfilled.
https://newrepublic.com/article/137410/pro-trump-intellectuals-want-overthrow-america
I would add one thing. Some of Strauss's most notable students, Stanley Rosen and Seth Bernadette for example, perhaps influenced by Socrates, stayed out of politics all together.