In this paper, our primary aim is to establish the extremizer stability of \(L^p\) -Rellich and Hardy–Rellich inequalities in the setting of Baouendi–Grushin vector fields. As a consequence, we obtain improved forms of these inequalities and derive an identity relating the subcritical and critical Hardy inequalities, thereby demonstrating their equivalence. These improvements are obtained through a suitably defined distance from extremizers. In the higher-order setting, we derive Hardy–Rellich type inequalities involving all radial operators in the Grushin framework and prove that all resulting constants are sharp. Finally, for the \(L^2\) -higher-order cases, we compute exact remainder terms by establishing identities rather than inequalities.
