Proposition 7

Let $g$ be a Lie algebra, $\beta$ an invariant symmetric bilinear form on $g$ and $a$ an ideal of $g$.

(a) The orthogonal $a'$ of $a$ with respect to $\beta$ is an ideal of $g$. (b) If $a$ is characteristic and $\beta$ is completely invariant, $a'$ is characteristic. (c) If $\beta$ is non-degenerate, $a \cap a'$ is commutative.

Let $D$ be a derivation of $g$. Suppose that $a$ is stable under $D$ and that $\beta(Dx, y) + \beta(x, Dy) = 0$ for $x, y$ in $g$. Then $z \in a'$ implies $Dz \in a'$, since, for all $t \in a$, $Dt \in a$ and hence $\beta(Dz, t) = -\beta(z, Dt) = 0$. Thus $a'$ is stable under $D$. This establishes (a) and (b).

Now let $b$ be an ideal of $g$ and suppose that the restriction of $\beta$ to $b$ is zero. For $x, y$ in $b$ and $z \in g$, $\beta([x, y], z) = \beta(x, [y, z]) = 0$, for $[y, z] \in b$. Thus $[b, b]$ is orthogonal to $g$. If $\beta$ is non-degenerate, $b$ is therefore commutative. This result applied to $a \cap a'$ proves (c).