The Laplacian measures the second order average difference of u in a neighbourhood around a point, i.e.
$$ \frac{1}{v(B_r)} \int_{B_r(x)} [f(y) - f(x)] = C \Delta u(x) r^2 + o(r^2) $$
Where the constant $C$ depends on the dimension. On the other hand, in a Lie group, the second order terms of the conjugate $ghg^{-1}h^{-1}$ forms the Lie bracket structure on the induced Lie algebra, which is a sort of 'difference measurement'. Is there a connection between these two 'second order' operators in mathematics?