Proposition 4

Consider the bilinear mapping $(u, v) \mapsto u * v$ (resp. $(u, v) \mapsto v * u$) of $U(G) \otimes K^{(G)}$ into $\mathscr{T}^{(\infty)}(G)$. The corresponding linear mapping of $U(G) \otimes K^{(G)}$ into $\mathscr{T}^{(\infty)}(G)$ is a vector space isomorphism.

$K^{(G)}$ is the direct sum of the $K_{e_x}$ for $x \in G$. On the other hand, the mapping $u \mapsto u * \varepsilon_g$ (resp. $u \mapsto \varepsilon_g * u$) is an isomorphism of the vector space $U(G) = \mathscr{T}_e^{(\infty)}(G)$ onto the vector space $\mathscr{T}_g^{(\infty)}(G)$ by Proposition 3. Finally, $\mathscr{T}^{(\infty)}(G)$ is the direct sum of the $T_g^{(\infty)}(G)$ for $g \in G$.

Let X be a manifold of class $C^r$ ($r \geq \infty$) and $x \in X$. We have defined (Differentiable and Analytic Manifolds, R, 13.3.1) a canonical filtration on the vector space $\mathscr{T}_x^{(\infty)}(X)$ and a canonical isomorphism $i_{X,x}$ of the associated graded vector space onto the graded vector space TS(T_x(X)). In particular, let $T_e(G) = L$; then $i_{G,e}$ is an isomorphism of the graded vector space gr U(G) onto the graded vector space TS(L). But U(G) is a filtered algebra, from which we obtain a graded algebra structure on gr U(G).