Hi, this is my first post here, so I hope I am asking the question the right way.

I am trying to understand to following piece of algebra: In his paper, Witten claims that $\int_M Tr(B \wedge DB) + \int_M Tr(\phi \wedge \ast D \ast B) = \langle B , \ast DB \rangle + \langle \phi, D \ast B \rangle$ (where B is a Lie-algebra valued 1-form, $\phi$ is a Lie-algebra valued 3-form, $\ast$ is the Hodge star, D is the covariant derivative with respect to some flat connection, and $M$ is a compact closed Riemannian 3-manifold) can be regarded as a product of the form $\langle H, L_- H \rangle$, where $H = B+\phi \in \Omega^1(M,\mathfrak{g}) \oplus \Omega^3(M,\mathfrak{g})$ and $L_- = \ast D + D \ast$ is what he calls the twisted Dirac Operator acting on 1 and 3 forms. The scalar product just comes from extending the inner products of 1- and 3-forms orthogonally onto the direct sum, i.e. 1-forms and 3-forms are orthogonal w.r.t. this inner product.

Witten does not bother to go into any detail explaining that, so I looked it up in another book, "Differential Topology and Quantum Field Theory" by Charles Nash. Now he claims the following (essentially equation 12.104):

$\langle H, L_- H\rangle = \langle B + \phi, (\ast D + D \ast) (B+\phi) \rangle = \langle B+\phi, \ast D B + D \ast B + D \ast \phi\rangle $ (the other term with $\phi$ drops out because $\phi$ is a 3-form, so $D\phi=0$) $= \langle B ,\ast D B\rangle + \langle B, D\ast \phi\rangle + \langle \phi, D \ast B \rangle$. So far so good, it's the linearity of the inner product and the fact that 1- and 3-forms are orthogonal to each other. Now he continues \begin{eqnarray} \langle H, L_- H \rangle = \int_M Tr (B \wedge DB) + \int_M Tr(B \wedge \ast D \ast \phi) + \int_M Tr(\phi \wedge \ast D \ast B) \end{eqnarray} \begin{eqnarray} = \int_M Tr(B\wedge DB) + 2 \int_M Tr(\phi \wedge D^\dagger B) \end{eqnarray} where $D^\dagger$ is the codifferential of $D$, i.e. $\langle \alpha, D\beta \rangle = \langle D^\dagger \alpha, \beta \rangle$ for differential forms $\alpha, \beta$ with the right degree. Now I do not see at all how he gets to the last expression. I don't mind the factor of 2, but I don't see how he manages to get the codifferential in this way. I have tried using Stokes as well as the definition of the codifferential and my calculations say that the last two terms in the first line should cancel. However I have to admit that I did not bother about the Lie-algebra part of the forms, i.e. I basically did it for the abelian case. But I was assured that it shouldn't matter. But apparently, it does...

I am pretty desperate to understand this part, so I would be happy about any kind of help you guys can offer me!