Exercise 26, § 6

Let $K$ be a field of characteristic $\neq 2$ and $\Phi$ a non-degenerate symmetric bilinear form on an $n$-dimensional vector space over $K$.

(a) Show that, if $n \geq 5$, the Lie algebra $\mathfrak{o}(\Phi)$ is simple. (By extending the base field, it can be reduced to the case where the index of $\Phi$ is $m = [n/2]$. If for example $n = 2m$ is even, $\mathfrak{o}(\Phi)$ has basis consisting of the elements

then argue as in Exercises 24 and 25. Proceed similarly when $n = 2m + 1$ is odd.)

(b) Let $\Delta$ be the discriminant of $\Phi$ with respect to a basis. Suppose that $n = 4$. Show that, if $\Delta$ is not a square in $K$, $\mathfrak{o}(\Phi)$ is simple. If $\Delta$ is a square, $\mathfrak{o}(\Phi)$ is the product of two isomorphic 3-dimensional simple Lie algebras. (Use the structure of $\mathbf{G}(\Phi)$ described in Algebra, Chapter IX, § 9, Exercise 16.) Deduce an example of a simple Lie algebra which becomes non-simple when extending the base field.