matrices - Area of a surface in terms of the densitized triad

Hi,
I need to know if this relation is correct for a metric:

$g_{a[b}g_{c]d}=frac{1}{2}epsilon_{ace}epsilon_{bdf}gg^{ef}$

I know that :

$frac{1}{2}epsilon_{ace}epsilon_{bdf}g^{ef}=g_{b[a}g_{c]d}$

but I don't see how the determinant $g$ of the metric could appear.

Edit:

Ok so the previous relation emerged when computing the area of a surface $S$ in terms of the "densitized" triad $E_{i}^{a}=ee_{i}^{a}$ where $a,b,c,...$ are the spatial coordinates and $i,j,k,...$ are $SU(2)$ coordinates, e the determinant of the triad matrix defined by $g_{ab}=e_{a}^{i}e_{b}^{j}delta_{ij}$ where $g_{ab}$ is the spatiale metric. So, since the computation of the area uses the determinant of the the metric $h_{alphabeta}$ induced by $g_{ab}$ on $S$: ($alpha,beta,... =1,2;and; a,b,..=1,2,3$)

$h_{alphabeta}=g_{ab}frac{partial x^{a}}{partialsigma^{alpha}}frac{partial x^{b}}{partialsigma^{beta}}$

So in computing the determinant $h$ explecitely on finds the term

$g_{a[b}g_{c]d}$ which needs to equal to $frac{1}{2}epsilon_{ace}epsilon_{bdf}gg^{ef}$
in order to obtain the final result:

$h=E_{i}^{a}E_{j}^{b}delta^{ij}n_{a}n_{b}$ where $n$ are normal vectors $n_{a}=epsilon_{abc}frac{partial x^{b}}{partialsigma^{1}}frac{partial x^{c}}{partialsigma^{2}}$

EDIT2:

After the notification of Willie Wong, I decided to put my original problem as a question, i.e: deriving the expression of the determinant of the induced metric on $S$ in terms of the densitized triad.