I'm using the reference "Differential Geometry, Gauge Theories and Gravity" by M. Göckeler and T. Schücker and I am having trouble to vary correctly the lagrangian
$$ \mathcal{L}_M=\dfrac{1}{2g^2}F \wedge *F $$
with respect to the vierbein $e^a$ in order to find the Energy-momentum of the Maxwell action.
By doing $e\rightarrow e+f$ in the lagragian above I find $$ \mathcal{L[e+f]}_M-\mathcal{L[e]}_M=\dfrac{1}{g^2}f^c\left(\frac{1}{4} F^{ab}\epsilon_{abcd} e^d \wedge F+\frac{1}{2}F_{cb}e^b \wedge *|_eF\right). $$
However, the right answer, present in the foregoing reference is
$$ \mathcal{L[e+f]}_M-\mathcal{L[e]}_M=\dfrac{1}{g^2}f^c\left(\frac{1}{4} F^{ab}\epsilon_{abcd} e^d \wedge F-\frac{1}{2}F_{cb}e^b \wedge *|_eF\right). $$
(the only difference is the minus sign in the expression inside the brackets)
I worked a lot, but couldn't identify my mistake. So, does anyone know anything that is not trivial in the treatment with the forms in this case?
(In this case the signature used is lorentzian)
The whole calculation that I did was the following:
Since
$$ \mathcal{L}_M[e]=\dfrac{1}{2g^2}F \wedge *F=\frac{1}{2g^2}\left[ \frac{1}{2}F_{ab} e^a \wedge e^b \wedge \left(\frac{1}{4}F^{\alpha \beta}\epsilon_{\alpha \beta cd}e^c \wedge e^d\right) \right] =\frac{1}{16g^2}\left( F_{ab}F^{\alpha \beta}\epsilon_{\alpha \beta cd} e^a \wedge e^b \wedge e^c \wedge e^d \right) $$
Then, by doing $e \rightarrow e+f$, and neglecting terms quadratic in $f$, we are lead to
$$ \mathcal{L}_M[e+f]-\mathcal{L}_M[e] =\frac{1}{16g^2} F_{ab}F^{\alpha \beta}\epsilon_{\alpha \beta cd} \left(f^a \wedge e^b \wedge e^c \wedge e^d + e^a \wedge f^b \wedge e^c \wedge e^d + e^a \wedge e^b \wedge f^c \wedge e^d + e^a \wedge e^b \wedge e^c \wedge f^d \right) =\frac{1}{16g^2} F_{ab}F^{\alpha \beta}\epsilon_{\alpha \beta cd} \left(2f^a \wedge e^b \wedge e^c \wedge e^d + 2e^a \wedge e^b \wedge f^c \wedge e^d \right) =\frac{1}{8g^2} F_{ab}F^{\alpha \beta}\epsilon_{\alpha \beta cd} \left(f^a \wedge e^b \wedge e^c \wedge e^d + e^a \wedge e^b \wedge f^c \wedge e^d \right) =\frac{1}{2g^2} \left(\frac{1}{4}F_{ab}F^{\alpha \beta}\epsilon_{\alpha \beta cd}f^a \wedge e^b \wedge e^c \wedge e^d + \frac{1}{4}F_{ab}F^{\alpha \beta}\epsilon_{\alpha \beta cd}e^a \wedge e^b \wedge f^c \wedge e^d \right) =\frac{1}{2g^2} \left[f^a \wedge \left(\frac{1}{4}F_{ab}F^{\alpha \beta}\epsilon_{\alpha \beta cd}\right) e^b \wedge e^c \wedge e^d + f^c \wedge \left(\frac{1}{4}F_{ab}F^{\alpha \beta}\epsilon_{\alpha \beta cd}\right) e^a \wedge e^b \wedge e^d \right] =\frac{1}{2g^2} \left[f^a \wedge \left(\frac{1}{4}F_{ab}F^{\alpha \beta}\epsilon_{\alpha \beta cd}\right) e^b \wedge e^c \wedge e^d + f^a \wedge \left(\frac{1}{4}F_{cb}F^{\alpha \beta}\epsilon_{\alpha \beta ad}\right) e^c \wedge e^b \wedge e^d \right] $$
And then, we have $$ \mathcal{L}_M[e+f]-\mathcal{L}_M[e] =\frac{1}{2g^2} f^a \wedge \left[ \left(\frac{1}{4}F_{ab}F^{\alpha \beta}\epsilon_{\alpha \beta cd}\right) e^b \wedge e^c \wedge e^d + \left(\frac{1}{4}F_{cb}F^{\alpha \beta}\epsilon_{\alpha \beta ad}\right) e^c \wedge e^b \wedge e^d \right] =\frac{1}{2g^2} f^a \wedge \left[ F_{ab}e^b \wedge *|_e F + \left(\frac{1}{2}F^{\alpha \beta}\epsilon_{\alpha \beta ad}\right) F \wedge e^d \right] =\frac{1}{2g^2} f^a \wedge \left[ F_{ab}e^b \wedge *|_e F + \left(\frac{1}{2}F^{\alpha \beta}\epsilon_{\alpha \beta ad}\right) e^d \wedge F \right] $$
Relabeling the indices, we finally have
$$ \mathcal{L}_M[e+f]-\mathcal{L}_M[e] =\frac{1}{g^2} f^c \wedge \left[\frac{1}{2} F_{cb}e^b \wedge *|_e F + \left(\frac{1}{4}F^{ab}\epsilon_{abcd}\right) e^d \wedge F \right] =\frac{1}{g^2} f^c \wedge \left[\left(\frac{1}{4}F^{ab}\epsilon_{abcd}\right) e^d \wedge F +\frac{1}{2} F_{cb}e^b \wedge *|_e F \right] $$
This is not in accordance with the reference due to the plus sign (it should be a minus) and, again, I could not identify where I have done something wrong.