Exercise 12, § 13

The Killing form of $\mathfrak{s}\mathfrak{l}(n, k)$ is $(X, Y) \rightarrow 2n$ Tr($XY$). That of $\mathfrak{s}\mathfrak{p}(n, k),n$ even, is $(X, Y) \rightarrow (n$+2)Tr($XY$). That of $\mathfrak{o}_S(n, k)$, where $S$ is non-degenerate symmetric of rank $n$, is $(X, Y) \rightarrow (n-$ 2)Tr($XY$).