Examples of minimal $G$-structures induced by the Lee form

We compute the condition of minimality of a G-structure for the Gray-Hervella class ${\mathcal{W}}_4$ of almost hermitian manifolds and ${\mathcal{C}}_5$ class of almost contact metric structures. We also consider ${\mathcal{C}}_4$ class by comparison with the Grey-Hervella class ${\mathcal{W}}_4$. The common feature is the existence of the Lee form $\theta$ representing these structures. We show that these classes contain minimal G-structures. Here, minimality means minimality of a $G$-structure inside oriented orthonormal frame bundle $SO(M)$ of a Riemannian manifold $M$.