I was reading Ch. 21 and I had some thoughts that may or may not be worth anything but i wanted to mention them to you.
1. the fact that all Execptional Lie Groups can be obtained from one equation is very interesting. Is there some connection between this equation and the Octonian intergers,and the Octonion Rings? See Conway and Smith"On Quaterions and Octonions"
2.The auotmorphism of the Octonians is G2. Therefore since the Octonions generate  the Exceptional Lie Groups shouldn't in principal the evauluation of buble diagrams in terms of  these  Execptional groups be reduciable to  the evaulation in terms of G2 which you have already done?
3.Have you seen Dixon's generation of the Magic Square by tensor products of the division algebras? See Geoffrey M. Dixon " Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics" Kluwer Academic Publishers. Note his derivation of the division algebrs from the Sayley-Dickson doubling prescription and independently from Galois sequences. Can one derive a asingle paramatised relation among the Lie groups from these procedures? I thing it should be possable. Would one get the same relation from the Galois procedure?
It seems like that in order to do mathematical physics that can yeild predictions about physical objects we  need to use an <alge><bra> (pardon the pun) which permits division and therefore we end up with the Lie Groups.
Prehaps because there ae only 4 such algebra that is why there ae only 4 infinite Lie
Groups  and 4 Execptional Lie Groups(F4,E6,E7, E8) with G2 being the underlying symetrey of these groups since it contains the  basis from which all others are built ( one open and one dark root)(seeGeorgi "Lie Algebras in Particle Physics" 2nd Ed.Section 20.1).
Prehaps this can be connected to the underlying topology. SeeFrankel " The Geometry of Physics" Ch.4, 15 and 21.4.
You may be wondering who I am. I I have a BS in physics from CCNY, MA in Astronomy from Harvard and a Ph.D in Earth and Space Sciences from SUNY_Stonybrook. I have not done any science for about the past 25 years or so and I have been practicing  law during that period. I have to support my family and I like it.
It would be nice to have someone to talk to about these things.
Joseph Kornblum