Exercise 18, § 6

(a) Let $g$ be a simple Lie algebra. Every invariant bilinear form on $g$ is either zero or non-degenerate. If $K$ is algebraically closed, every invariant bilinear form $\beta$ on $g$ is proportional to the Killing form $\beta_0$. (Consider the endomorphism $\sigma$ of the vector space $g$ defined by $\beta(x, y) = \beta_0(\sigma x, y)$ and show that $\sigma$ commutes with $\mathrm{ad}_g x$ and is hence scalar.) Show that this result may be false if $K$ is not algebraically closed. (Use Exercise 17 and the fact that the dimension of the space of invariant bilinear forms does not change when the base field is extended.)

(b) Let $g$ be a semi-simple Lie algebra over an algebraically closed field. Deduce from (a) and Exercise 7 (b) that the dimension of the space of invariant bilinear forms on $g$ is equal to the number of simple components of $g$ and that all these forms are symmetric.

(c) Let $g$ be a simple Lie algebra, $M$ the dual space of the vector space $g$, with the dual representation of the adjoint representation, and $h$ the semi-direct product of $g$ and $M$ defined by $x \mapsto x_M$. For $y, z$ in $g$ and $y', z'$ in $M$, we write $\beta(y + y', z + z') = \langle y, z' \rangle + \langle z, y' \rangle$. Show that $\beta$ is a non-degenerate invariant symmetric bilinear form on $h$, which is associated with no representation of $h$. (Observe that the radical and nilpotent radical of $h$ are equal to $M$.)