  Reservoir engineering has proven to be a practical approach to control open
quantum systems, preserving quantum coherence by appropriately manipulating the
reservoir and system-reservoir interactions. In this context, for systems
comprised of different parts, it is common to describe the dynamics of a
subsystem of interest by performing an adiabatic elimination of the remaining
components of the system. This procedure often leads to an effective master
equation for the subsystem that is not in the diagonal form of the
Gorini-Kossakowski-Lindblad-Sudarshan master equation (here called diagonal
Lindblad form). Instead, it has a more general structure (here called
non-diagonal Lindblad form), which explicitly reveals the dissipative coupling
between the various components of the subsystem. In this work, we present a set
of dynamical equations for the first and second moments of the canonical
variables for linear Gaussian systems, bosonic and fermionic, described by
non-diagonal Lindblad master equations. Our method is efficient and allows one
to obtain analytical solutions for the steady state. We supplement our findings
with a review of covariance matrix methods, focusing on those related to the
measurement of entanglement. Notably, our exploration yields a surprising
byproduct: the Duan criterion, commonly applied to bosonic systems for
verification of entanglement, is found to be equally valid for fermionic
systems. We conclude with a practical example, where we revisit two-mode
mechanical entanglement in an optomechanical setup. Our approach, which employs
adiabatic elimination for systems governed by time-dependent Hamiltonians,
opens the door to examine physical regimes that have not been explored before.
