Corollary 3

Let $\mathfrak{g}$ be a Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$, and let X be a subset of $\mathfrak{g}$ generating $\mathfrak{g}($as a Lie algebra over k).

(i) The decomposable envelope $e(\mathfrak{g})$ of $\mathfrak{g}$ is generated by the semi-simple and nilpotent components of the elements of X.

(ii) If $k'$ is an extension of $k,e(\mathfrak{g}\otimes_kk') =e(\mathfrak{g})\otimes_kk'$; and $\mathfrak{g}$ is decomposable if and only if $\mathfrak{g}\otimes_kk'$ is decomposable.

Let $\widetilde{\mathfrak{g}}$ be the subalgebra of $\mathfrak{g}\mathfrak{l}(V)$ generated by the semi-simple and nilpotent components of the elements of X. Then $\mathfrak{g}\subset \widetilde{\mathfrak{g}}\subset e(\mathfrak{g})$; by Th. $1, \widetilde{\mathfrak{g}}$ is decomposable, so $\widetilde{\mathfrak{g}}=e(\mathfrak{g})$, which proves (i). Assertion (ii) follows, since X generates the $k'$-algebra $\mathfrak{g}\otimes_kk'$.