= Solution
Use a dry <ideal gas> of fixed composition with specific gas constant $\mathcal R=C_p-C_v$ and constant <specific heat capacity at constant pressure> $C_p$. For a fixed-mass parcel, $PV^\gamma=$ constant and $PV=m_{\rm parcel}\mathcal RT_g$ imply
$$
T_gP^{-(\gamma-1)/\gamma}=\mathrm{constant},\qquad \frac{dT_g}{dP}=\frac{\mathcal R}{C_p}\frac{T_g}{P}=\frac1{\rho_gC_p}.
$$
Along a hydrostatic adiabat, $dP/dz=-\rho_gg$; hence the <dry adiabatic lapse rate> is
$$
\boxed{\left(\frac{dT_g}{dz}\right)_{\rm ad}=-\frac g{C_p}.}
$$
For an actual pressure-balanced parcel rising in an ambient atmosphere, $dP/dz=-\rho_{\rm env}g$ instead gives $dT_g/dz=-(g/C_p)(T_g/T_{\rm env})$. The usual lapse-rate expression is exact for a hydrostatic adiabatic column and is the local first-order result at the launch point where $T_g=T_{\rm env}$, as needed in a linear stability test. Treating an already much hotter parcel as an exact copy of the ambient hydrostatic column would be an extra approximation.
After a small upward displacement $\delta z$ from temperature equilibrium, its temperature excess is
$$
T_g-T_{\rm env}\simeq\left[-\frac g{C_p}-\frac{dT_{\rm env}}{dz}\right]\delta z.
$$
At equal pressure, warmer gas is less dense and continues to rise. Thus the <Schwarzschild criterion> in altitude form is
$$
\boxed{\frac{dT_{\rm env}}{dz}<-\frac g{C_p}\quad\text{unstable},\qquad \frac{dT_{\rm env}}{dz}=-\frac g{C_p}\quad\text{neutral}.}
$$
The supplied non-strict inequality includes the marginal case; strict growth requires the strict inequality. A downward displacement gives the same stability conclusion. Efficient <convection> normally adjusts an initially superadiabatic gradient to a nearly adiabatic one.
Deep envelopes of <gas giants> and <ice giants> commonly transport intrinsic heat by <convection>, as do the <planetary tropospheres> of many weakly irradiated atmospheres. <Earth>'s dry <troposphere> provides another approximate example, with moisture changing the lapse rate. Strongly irradiated <hot Jupiters> can still have deep convective interiors, while their upper radiative regions need not be convective. Composition gradients can modify the homogeneous-gas criterion and inhibit overturning even in an interior.
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