I am working through the math in Guo/Zheng/Shi paper. Something that is causing me trouble is going from equation 6 to equation 7. With the double dot product, is seems to me like the moments of F do not include all the terms. I would expect a Ccs^2I term for the zeroth moment, but that is not the result they obtained. Maybe I am not understanding the double dot product as I much as I should. Any help or hints will be greatly appreciated!

perhaps you have already figured this out but just in case.
taking the zeroth moment of Fi gives: ( . is dot product, : is tensor contraction, I is identitiy matrix)

from definitions of the lattice for d2q9: sum(wi) = 1, sum(wiei) = 0 and sum(wieiei) = cs^2I
this implies that the term sum(wieiei)-cs^2*I = 0
and for the zeroth moment follows: sum(Fi) = A as derived in the paper.