Garrett Lisi on Nostr: I don't know how much it will help, but I think I see how this works algebraically in ...

I don't know how much it will help, but I think I see how this works algebraically in 𝑓₄. One has the decompositions
𝑓₄ = 𝑠𝑜(9) + 16ˢ
and
𝑓₄ = 𝑠𝑢(3) + 𝑠𝑢(3) + 3×6 + 3̅×6̅
The 𝑠𝑢(3) + 𝑠𝑢(3) almost but doesn't quite fit in the 𝑠𝑜(9). The 𝑠𝑜(9) is
𝑠𝑜(9) = 𝑠𝑜(3) + 𝑠𝑜(6) + 3×6
= 𝑠𝑢(2) + 𝑠𝑢(4) + 3×6
= 𝑠𝑢(2) + 𝑠𝑢(3) + 𝑢(1) + 3 + 3̅ + 3×6
So you can see the ''intersection'' 𝑠𝑢(2)+𝑠𝑢(3)+𝑢(1) in 𝑠𝑜(9), and the related 𝑠𝑢(3)+𝑠𝑢(3) in 𝑓₄ is found by pulling a 2+2̅ out of the 16ˢ to make the
𝑠𝑢(3) = 𝑠𝑢(2) + 𝑢(1) + 2 + 2̅
with the other 𝑠𝑢(3) the same as in 𝑠𝑜(9).

Now, my thinking as to why this isn't especially great for Standard Model building is that 𝑢(1) in the 𝑠𝑜(9) acting on the 16ˢ is B-L and not hypercharge, Y. What does work well with the SM is this same game in 𝑒₆, with 𝑠𝑜(10) -- as you and JH figured out.