  We study the regularity properties of a general second order H\"ormander
operator with Dini continous coefficients $a_{ij}$. Precisely if $X_0,
X_1,\cdots X_m$ are smooth self adjoint vector fields satisfying the
H\"ormander condition, we consider the linear operator in $\mathbb{R}^{N}$,
with $N>m+1$: \begin{equation*} \mathcal{L} u := \sum_{i, j= 1}^{m} a_{ij}
X_{i}X_{j} u - X_0 u. \end{equation*} The vector field $X_0$ plays a role
similar to the time derivative in a parabolic problem so that it is a vector of
degree two. We prove that, if $f$ is a Dini continuous function, then the
second order derivatives of the solution $u$ to the equation $\mathcal{L} u =
f$ are Dini continuous functions as well. A key step in our proof is a Taylor
formula in this anisotropic setting, that we establish under minimal regularity
assumptions.
