  Fermionic Gaussian states have garnered considerable attention due to their
intriguing properties, most notably Wick's theorem. Expanding upon the work of
Balian and Brezin, who generalized properties of fermionic Gaussian operators
and states, we further extend their findings to incorporate Gaussian operators
with a linear component. Leveraging a technique introduced by Colpa, we
streamline the analysis and present a comprehensive extension of the
Balian-Brezin decomposition (BBD) to encompass exponentials involving linear
terms. Furthermore, we introduce Gaussian states featuring a linear part and
derive corresponding overlap formulas. Additionally, we generalize Wick's
theorem to encompass scenarios involving linear terms, facilitating the
expression of generic expectation values in relation to one and two-point
correlation functions. We also provide a brief commentary on the applicability
of the BB decomposition in addressing the BCH (Zassenhaus) formulas within the
$\mathfrak{so}(N)$ Lie algebra.
