Corollary 4

Let $g$ be a Lie algebra, $a$ an ideal of $g$ and $\rho$ a semi-simple representation of $g$. (a) The restriction $\rho'$ of $\rho$ to $a$ is semi-simple. (b) If $\rho$ is simple, $\rho'$ is a sum of simple representations isomorphic to one another.

Passing to the quotient by the kernel of $\rho$, $\rho$ can be assumed to be faithful. Then $g$ is reductive. Let $g = g_1 \times g_2$, where $g_1$ is the centre of $g$ and $g_2$ is semi-simple. Then $a = a_1 \times a_2$, where $a_1 \subset g_1$, $a_2 \subset g_2$ and $a_1$ is the centre of $a$. The elements of $\rho(g_1)$, and in particular those of $\rho(a_1)$, are semi-simple (Theorem 4) and hence $\rho'$ is semi-simple (Theorem 4). Hence (a). Assertion (b) follows from (a), using § 3, no. 1, Corollary to Proposition 1.