dg.differential geometry - A question about a formula of Pfaffian

The supertrace can be evaluated either by Berezin Gaussian integration, or equivalently by
summation over a Clifford algebra. Here is a description of the second method.

Let $omega$ be a skew-symmetric 2n by 2n matrix. Let ${{ e_1, e_2, . . . . e_{2n} }}$ be a real
2n-dimensional Clifford algebra. Then:

$exp(Sigma_{k,l=1}^{2n} omega_{kl} e_k e_l) = Sigma_{|K| even} Pf(omega_K) hat{e}_K$

where: $K$ is a subset of ${{ 1, 2, . . . . 2n }}$ and $hat{e}_K$ is the corresponding wedge product of the Clifford generators and $omega_K$ is the skew-symmeterized submatrix containg only the rows and columns in $K$.

Now, the supertrace selects the coefficient of the top form, giving you the required formula.

A proof of this result can be found for example in the following book by: José Gracia Bondía, Joseph C. Várilly, Héctor Figueroa.