Proposition 8

Let $V$ be a finite-dimensional vector space. Then $\mathrm{gl}(V)$ is reductive. Its centre is the set of homotheties of $V$, its derived algebra is $\mathrm{sl}(V)$ and the latter is semi-simple.

The identity representation of $\mathrm{gl}(V)$ is simple, hence $\mathrm{gl}(V)$ is reductive and therefore $\mathrm{gl}(V)$ is the direct sum of its centre $c$ and its derived algebra $\mathscr{D}(\mathrm{gl}(V))$. The centre $c$ is the set of homotheties (Algebra, Chapter II, § 2, no. 5, Corollary 1 to Proposition 5). Clearly $\mathscr{D}(\mathrm{gl}(V)) \subset \mathrm{sl}(V)$. As $\mathrm{sl}(V) \cap c = \{0\}$, $\mathscr{D}(\mathrm{gl}(V)) = \mathrm{sl}(V)$. Hence $\mathrm{sl}(V)$ is semi-simple.