A constant feature of Eliot's poetry, from INVENTIONS OF THE MARCH HARE 
through "Four Quartets" is images of the double.  Sometimes this is
explicit, as in the original image of the old man sitting in the gutter
in the omitted section of "Prufrock" or the familiar compound ghost who
is described as another self-- "I was still the same/ Knowing myself yet
being someone other--."  Sometimes it is implicit, as in the old man
waiter in "Dans le Restaurant" who dares to have the same feelings as
the appalled narrator.

At the time Eliot was working on TWL, during the last year of the War
when the US had entered, he was also trying over and over to enlist but
was never accepted because of a hernia.  He wanted to get into
intelligence work but failed at that.  The image of someone who was not
at the hot gates (Thermopylae) and did not fight precedes, in
"Gerontion" this accosting of one who was with me at the Battle of Mylae
but now is suspect and accused and is his double. 

If one wants to find meanings in allusions, one is going back to sources
in Eliot's reading and imagination, but there is as constant a
self-reference in his work as a text reference.  I think in this case
the hypocrite reader, double, brother is a projection of himself and
sense of guilt, as much an image of the terrible anxiety war caused him
as anything else.  I remain consistently puzzled at the assumption that
all of literature and every idea in history could be the source of
anything Eliot ever wrote--everything except what he experienced
himself.  Yet there it is over and over, as so much recent knowledge now
reveals.  Hugh Kenner, interestingly, mainly displaced both and focused
on a claim that what mattered to Eliot was effects, and that reading the
sources was not important.  At that time neither was biography
available.  But unlike many readers, he paid little attention to the
endless references from other sources.  I think there is still not a
really compelling theory of what part they played or how important they

About Jewel Brooker, her article on Eliot and the War looks at the way
he was affected and how it impacted his poetry.  I am not sure of the
reference but it was in Modernism/Modernity and could be found easily.