# Simple Lie algebra question

I am currently trying to up my understanding of Lie algebras as the brief introductions I have had from various QFT textbooks feels insufficient, but have been stuck on one small point for a couple days now. I am reading through the lecture notes / book by Robert Cahn found here: http://theory.uchicago.edu/~sethi/Teaching/P385-W2011/texall.pdf. On page 97, I am having trouble understanding how equation XI.31 comes about. I think I get how $Tr C = N_\Lambda \langle \Lambda, \Lambda + 2 \delta \rangle_2$ comes about, but don't see how I can use the statement regarding (( , )) yielding $l_\phi ( , )_2$ and how that directly leads to XI.31.

A bit of guidance here would be very much appreciated. Thanks.