In this paper I construct a parametrix for the hypoelliptic diffusion equations ( ∂ / ∂ t − L ) u = 0 (\partial /\partial t - L)u = 0 , where L = ∑ a = 1 n g a 2 L = \sum \nolimits _{a = 1}^n {g_a^2} and where the g a {g_a} are vector fields which satisfy the property that they, together with all of the commutators [ g a , g b ] [{g_{a,}}{g_b}] for a > b a > b , are at each point linearly independent and span the tangent space.