The aim of this paper is to study the following differential inequality \[ Hu(z)= \sum^m_{i,j=1} a_{ij}(z) Z_i Z_j u(z)+ Z_0 u(z)\geq 0,\quad z\in\Omega\subset \mathbb{R}^N,\tag{1} \] where \((a_{ij})_{i,j}\) is positive semidefinite, \(Z_0,Z_1,\dots, Z_m\) are locally Lipschitz-continuous vector fields on the open set \(\Omega\) and \(u\) is a real-valued function which needs to be differentiable. The authors are interested in weak maximum principles and maximum propagation for functions \(u\) in an intrinsic class of regularity modelled on the fields \(Z_j\), say \(u\in \Gamma^2(\Omega)\), a somewhat minimal regularity class for which inequality (1) makes sense. By \(\Gamma^2(\Omega)\), that is \(u\in\Gamma^2(\Omega)\), the authors mean that \(u:\Omega\to \mathbb{R}\) is a continuous function with continuous Lie-derivatives along \(Z_1,\dots, Z_m\) up to the second-order and a continuous Lie-derivative along \(Z_0\).