Help.. differntial forms

[DanM]

I need to compute the following thing

trace ( (dg + g^g)^2) = d (tr (g ^ dg + 2/3 g ^ g ^ g))

im messing up completely b/c I dont know how to use trace with differential forms

tr (dg + g^g) ^ (dg + g^g))

[Haelfix]

Just expand it out, theres only a few tricky problems

1) This is the Chern-Simons form, so you are dealing with Matrix valued forms.. So in general

tr (A ^ B) = -(1)^(rs) tr (B ^ A) (err 2nd wedge is a power)

Use the Bianchi identities

and of course

d (A ^ B) = dA^B + (-1)^rs (A^dB)

Note that
tr (A^A^A^A) = 0 b/c of the last identity

Expand out both sides, it should only take you a few lines.