Exercise 7, § 5

Let $\mathfrak{g}$ be a Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$. Let $m, n$ be two integers $\geq 0, W$ and $W'$ two vector subspaces of $\mathbf{T}^m(V)\otimes \mathbf{T}^n(V^*)$ where $V^*$ is the dual of V. Assume that $W'\subset W$ and that W and $W'$ are stable under the natural representation of $\mathfrak{g}$ on $\mathbf{T}^m(V)\otimes \mathbf{T}^n(V^*)$. Show that W and $W'$ are then stable under $e(\mathfrak{g})$. If $\pi$ denotes the representation of $e(\mathfrak{g})$ on $W/W'$ thus obtained, show that $\pi e(\mathfrak{g})$ is the decomposable envelope of $\pi (\mathfrak{g})$ (use Th. 1).

Deduce that ad $e(\mathfrak{g})$ is the decomposable envelope of ad$\mathfrak{g}$ in $\mathfrak{g}\mathfrak{l}(\mathfrak{g})$.

$^8$ For more details, cf. C. CHEVALLEY, Théorie des groupes de Lie, II, Groupes algébriques, Chap. II, §14, Paris, Hermann, 1951.