Lemma 3

Let g be a semi-simple Lie algebra, $\rho$ a linear representation of g on a finite-dimensional vector space V and W a subspace of V of codimension 1 such that $\rho(x)(V) \subset W$ for all $x \in g$. Then there exists a supplementary line of W which is stable under $\rho(g)$.

For all $x \in g$ let $\sigma(x)$ be the restriction of $\rho(x)$ to W. Suppose first that $\sigma$ is simple. If $\sigma = 0$, then $\rho(x)\rho(y) = 0$ for all $x, y$ in $g$, hence $\rho(g) = \rho(\mathscr{D}g) = \{0\}$ and our assertion is obvious. If $\sigma \neq 0$, let n be the kernel of $\sigma$ and let m be a supplementary ideal of n in $g$ (Lemma 1); then $m \neq \{0\}$ and the restriction of $\sigma$ to m is faithful; the restriction to m of the bilinear form associated with $\sigma$ is non-degenerate (Proposition 1) and hence the Casimir element c associated with m and $\sigma$ can be formed. By Proposition 12 of § 3, no. 7, $\sigma(c)$ is an automorphism of W. On the other hand, $\rho(c)(V) \subset W$. Hence the kernel Z of $\rho(c)$ is a supplementary line of W; since c belongs to the centre of the enveloping algebra of $g$, $\rho(c)$ is permutable with $\rho(x)$ for all $x \in g$ and hence Z is stable under $\rho(g)$.

In the general case we argue by induction on the dimension of V. Let T be a minimal non-zero stable subspace of W. Let $\rho'$ be the quotient representation on $V' = V/T$. Then, for all $x \in g$, $\rho'(x)(V') \subset W'$, where $W' = W/T$ is of codimension 1 in $V'$. By the induction hypothesis there exists a line $Z'$ which is supplementary to $W'$ and stable under $\rho'(g)$. Its inverse image Z in V is stable under $\rho(g)$, contains T as subspace of codimension 1, $Z \cap W = T$, and hence $\rho(x)(Z) \subset T$ for all $x \in g$. By what was proved above, there exists a supplementary line of T in Z which is stable under $\rho(g)$; this line is supplementary to W in V, which completes the proof.