We identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold. 1. Introduction The space of smooth ﬁrst order linear diﬀerential operators on Rn that preserve harmonic functions is closed under Lie bracket. For n ≥ 3, it is ﬁnitedimensional (of dimension (n2 + 3n + 4)/2). Its commutator subalgebra is isomorphic to so(n + 1, 1), the Lie algebra of conformal motions of Rn . Second order symmetries of the Laplacian.