Exercise 20, § 6

Suppose that K is algebraically closed. Let $g$ be a Lie algebra and c its centre.

(a) If $g$ admits a finite-dimensional faithful simple representation, $g$ is reductive and $\dim c \leq 1$.

(b) Conversely, if $g$ is reductive and $\dim c \leq 1$, $g$ admits a finite-dimensional faithful simple representation. (If $g$ is simple, use the adjoint representation. If $g = g_1 \times g_2$ and $g_1, g_2$ admit finite-dimensional faithful simple representations $\rho_1, \rho_2$ on spaces $M_1, M_2$, show that

is a semi-simple representation $\rho$ of $g$, then that the centralizer of the $g$-module $M_1 \otimes M_2$ reduces to the scalars and hence that $\rho$ is simple. When $g_1$ and $g_2$ are semi-simple or $g_1$ is semi-simple and $g_2$ is commutative, show that $\rho$ is faithful by considering the kernel of $\rho$ which is an ideal of $g_1 \times g_2$.)