Exercise 21, § 6

Let V be a vector space over K of finite dimension $n > 1$. Show that $\mathfrak{sl}(V)$ is simple. (Reduce it to the case where K is algebraically closed. Suppose that $\mathfrak{sl}(V) = a \times b$, with $\dim a = a > 0, \dim b = b > 0$. Let A (resp. B) be the associative algebra generated by 1 and a (resp. b). Then V can be considered as a simple ($A \otimes_K B$)-module. Hence there exists an A-module P of finite dimension $p$ over K and a B-module Q of finite dimension $q$ over K such that V is ($A \otimes_K B$)-isomorphic to $P \otimes_K Q$ (Algebra, Chapter VIII, § 7, no. 7, Proposition 8 and no. 4, Theorem 2); P and Q are faithful and hence $a \leq p^2, b \leq q^2$. On the other hand, $a + b = n^2 - 1$ and $pq = n$, whence $(p^2 - 1)(q^2 - 1) \leq 2$, which is contradictory.)