Exercise 2, § 5

Let G be a group of class $C^r$ with $r \geq 2$. (a) Let $t \in \mathscr{T}^{(s)}(G)$, $t' \in \mathscr{T}^{(s')}(G)$. If $s + s' \leq r$, define $t * t' \in \mathscr{T}^{(s+s')}(G)$ as in § 3, no. 1. If t and $t'$ are without constant term, $t * t'$ is without constant term. The image of t under the mapping $x \mapsto x^{-1}$ of G into G is denoted by $t^\vee$; then $t^\vee \in \mathscr{T}^{(s)}(G)$. If $f$ is a function of class $C^r$ on G with values in a Hausdorff polynormed space, $f * t$ denotes the function on G defined by $(f * t)(x) = \langle \varepsilon_x * t^\vee, f \rangle$ for all $x \in G$; this function is of class $C^{r-s}$ if $s < \infty$. Show, as in § 3, no. 4, that $f * (t * t') = (f * t) * t'$. (b) If $t \in T_e(G)$, the vector field $x \mapsto \varepsilon_x * t$ on G is denoted by $L_t$. Show, as in § 3, no. 6, that, for $t, t'$ in $T_e(G)$, $L_{t*t'} = L_t \circ L_{t'}$ and hence that

Therefore, $t * t' - t' * t$ is an element of $T_e(G)$ which we denote by $[t, t']$. (c) Using the notation of Exercise 1 (a), if $t \in T_e(G)$, then for $x$ sufficiently close to 0. Deduce that $[t, t'] = B(t, t') - B(t', t)$. (d) Show that $T_e(G)$, with the bracket $(t, t') \mapsto [t, t']$, is a normable Lie algebra. (To prove the Jacobi identity, use (c), Exercise 1 (b) and identity (5) of Algebra, Chapter I, § 6, no. 2.) (For a sequel to this exercise, cf. § 8, Exercise 6.)