In the wake of scandals involving alleged antisemitic statements said to have been uttered by sundry Labour stalwarts, Jeremy Corbyn has announced the establishment of an "internal inquiry" into these matters. What, realistically, are the chances of this investigation leading to real change?

Don't be fooled into thinking Labour's lacklustre electoral performance last week must have added any urgency to this investigation. True, Labour is the first Opposition in decades to have lost seats mid-term. True, Labour was all but wiped out in Scotland. But, in England, Labour did not do nearly as badly as some predicted. And Sadiq Khan was elected to the London mayoralty.

As London's first Muslim mayor Khan - who, at one time, was happy to associate himself with the BDS movement - is very much on probation. When, in the now infamous BBC Radio interview with Vanessa Feltz, Ken Livingstone claimed that Hitler supported Zionism and that he (Livingstone, a Labour party member for 47 years) had "never heard anyone [within Labour] saying anything antisemitic," Khan was quick to denounce these comments.

I agree with Khan that Red Ken's comments were indeed "appalling and inexcusable." But don't expect any apology. Speaking on a London-based Arabic language TV station (Al-Ghad Al-Arabi) on May 4, Livingstone was unrepentant, denouncing the creation of the state of Israel as "a great catastrophe" and "fundamentally wrong."

But we owe Livingstone a debt of gratitude. In the course of his interview with Feltz, he unburdened himself of a conviction that he clearly holds very dear: namely that it was wrong (and I quote) "to think of antisemitism and racism as exactly the same thing." To Livingstone, and to many other Labour party members, there's racism, which is worthy of condemnation, and then there's antisemitism. By what might be termed the acceptable face of antisemitism, these comrades mean criticism of Israel and characterising its creation (to quote Corbyn's director of strategy, Seumas Milne, speaking in 2009) as "this crime."

I have no doubt that the Labour file will be quietly shut

In reminding us that, in his view, there's a fundamental difference between racism and anti-Jewish prejudice of this sort, Livingstone has surely set out the core agenda for the internal inquiry that Corbyn has been forced to concede. Its precise terms of reference have yet to be announced. But we know that in January 2015 its deputy chair, professor David Feldman, presented to the Parliamentary Committee Against Antisemitism a report on "antisemitism in public debate … with particular reference to allusions to the Holocaust, to the idea that Jews constitute a body in British society which puts Israel's interests first and to the notion that there are 'good Jews and 'bad Jews'." What is valuable in this document is not so much Feldman's analysis of public debate but rather the definitions of antisemitism he deploys. One of these "malign stereotypes" is the idea that Jews constitute a cohesive community fixated on the pursuit of its own sectarian ambitions. Another is grounded in "discriminatory practices which disadvantage Jews."

Feldman has no difficulty in condemning such practices as "antisemitic." But he is adamant that "the application of double standards" (singling out Israel for criticism while ignoring totalitarian regimes) "does not itself constitute antisemitism" - though it may do if underpinned by "a malign stereotype of Jews applied to the state of Israel." Moreover, he explicitly rejects (as do I) the definition widely enshrined in the Macpherson inquiry into the murder of black teenager Stephen Lawrence, that "a racist incident" is simply any incident "perceived to be racist by the victim or any other person."

I expect Corbyn's "internal inquiry" to be narrowly focused, and to be rooted firmly in Feldman's theoretical approach. For example, Feldman is clear that allusions to Hitler and comparisons between Israel and the Nazi state, though they may well fall into one or other of his categories of antisemitism, will not automatically do so, no matter how offensive Jews may find them. But - and it is a big but - Feldman also writes that "we should all exercise restraint in our own recourse to analogies to the Holocaust and the Nazi era."

Some of the central characters in the recent drama within the Labour party are clearly incapable of exercising such restraint. Their token expulsion from the party may well follow. And then the file will be quietly closed.