Let \(M\) be a sub-Riemannian manifold. Suppose that the Hörmander condition holds. Then to each point \(p\in M\) we can associate its degree of nonholonomy \(r(p)\) which counts how many bracket iterations of horizontal vector fields near \(p\) are needed to span the tangent space \(T_pM\). The point is called regular if \(r\) is constant near \(p\), singular otherwise. Using the notion of pointed Gromov-Hausdorff limit one can make \(T_pM\) into a metric space. It is shown that \(T_pM\) with this metric is itself a sub-Riemannian manifold. The proof uses special adapted coordinates. If \(p\) is regular, then \(T_pM\) naturally carries the structure of a nilpotent Lie group. In the Riemannian case, the Lie group structure is given by addition, hence it is abelian. If \(p\) is singular, then \(T_pM\) is a homogeneous space of a simply connected nilpotent Lie group by a connected subgroup.